Nonequivalent definitions in Mathematics I would like to ask if anyone could share any specific experiences of 
discovering nonequivalent definitions in their field of mathematical research. 
By that I mean discovering that in different places in the literature, the same name  is used for two different mathematical objects.
 This can happen when the mathematical literature grows quickly and becomes chaotic and of course this could be a source of serious errors.
I have heard of such complaints by colleagues,  mostly with respect to definitions of various spaces and operators, but I do not recall the specific (and very specialised) examples.
My reason for asking is that I'm currently experimenting with verification using proof assistants and I'd like to test some cases that might be a source of future errors.
EDIT: Obviously, I would be interested to see as many instances of this issue as possible so I would like to ask for more answers. Also, it would be even more helpful if you could provide references.
 A: An $n$-category used to mean a strict $n$-category (a category enriched in the cartesian closed category $(n-1)$-$\mathsf{Cat}$), and now is often used to mean a weak $n$-category (which itself is defined in multiple ways). 
I'd like to mention also that many of the examples in the answers and comments are examples of what the nLab calls red herrings, which need be neither red nor herrings, or sometimes all herrings are red herrings. For example, a multi-valued function is not actually a function, but all functions are multi-valued functions. A manifold with boundary is not actually a manifold, but all manifolds fit the definition of manifold with boundary, hence the locution "manifold without boundary". The category of non-associative algebras includes associative algebras. 
A: *

*The category of models of a (finitary or infinitary) first-order theory

*A model category which is an abstract setting for doing homotopy theory


The first notion is expounded for example in the Adamek and Rosicky book "Locally Presentable and Accessible categories", chapter 5. As for the second notion, a possible starting point is https://ncatlab.org/nlab/show/model+category.
A: Is a parallelogram also a trapezoid?
A: Several distributions from probability theory share names:
The log-gamma distribution. Similarly to the log-normal distribution where we say $X$ is log-normal if $Y = \log X$ is normally (Gaussian) distributed, we can say that $A$ is log-gamma if $B = \log A$ is gamma distributed.
But I have also seen $A$ being called log-gamma if $B = \exp A$ is gamma distributed.

The Weibull distribution may refer to the heavy-tailed distribution function or one of the three max-stable distributions.

Another example is given by Mittag-Leffler distributions, which are distributions on $\mathbb{R}_+$. For $\alpha \in (0,1]$, let $E_\alpha (z) := \sum_{n \geq 0} \Gamma (1+n\alpha)^{-1} z^n$ be
the Mittag-Leffler function of index $\alpha$. Then a (normalized) Mittag-Leffler distribution may refer to:


*

*a distribution whose CDF is defined with the Mittag-Leffler function:


$$\mathbb{P} (X_\alpha \geq t) = 1-E_\alpha (-t^ \alpha) \quad \forall t \geq 0;$$
these are heavy-tailed distributions, with Laplace transform $(1+\lambda^ \alpha)$, and used mostly in statistics, economics, etc.


*

*a distribution whose MGF is defined with the Mittag-Leffler function:


$$\mathbb{E} (e^{zX_\alpha}) = E_\alpha (z) \quad \forall z \in \mathbb{C};$$
their densities decay quickly at infinity, and they appear as the limit of local times of Markov processes.
A: One of the most classical examples is the word «algebra», which denotes not only a branch of mathematics, but also the following mathematical objects:
In linear algebra an algebra is a vector space, equipped with a bilinear operator (called product). 
In set theory an algebra is a collection of sets closed under finite union, finite intersection and complement.
In universal algebra an algebra is a set, equipped with collection of finitary operations.
Note, that any algebra in linear algebraic sense is also an algebra in universal algebraic set, but not vice versa.
A quite similar thing happens with varieties:
In algebraic geometry a variety is the set of solutions of a system of algebraic equations.
In universal algebra a variety is a class of all algebras (in universal algebraic sense) with a given signature, satisfying a given set of identities.
Moreover, both those «varieties» are translated to Russian as «многообразия» - the same word, that is used for manifolds (topological spaces, such that each point of them has a neighbourhood, that is homeomorphic to $\mathbb{R}^n$ for some fixed $n$)
And if I have reached the theme of ambiguous translations, I think, that I should mention that «perfect groups» (groups equal to their derived subgroup), «complete groups» (centerless groups isomorphic to their automorphism group) and «immaculate groups» (finite groups, whose order is equal to the sum of orders of their proper normal subgroups) are all translated to Russian as «совершенные группы».
Also, the following examples deserve to be mentioned:
Two abstract groups are called commensurate (or commensurable), if they have isomorphic subgroups of finite index. Two subgroups of a group are called commensurate (or commensurable) if their intersection has finite index in both of them. Note, that two subgroups of a group may be commensurate as abstract groups, but not commensurate as subgroups.
Artinian groups are groups, that satisfy the minimum condition on subgroups. Artin groups are groups, that have a presentation of specific form. Both of them are «Артиновы группы» in Russian.
A right (left) ideal of a ring is a subring, that is closed under right (left) multiplication on arbitrary element of the ring. A right (left) ideal of a semigroup is a subsemigroup, that is closed under right (left) multiplication on arbitrary element of the semigroup. Note, that an ideal of the multiplicative semigroup of an associative ring is not always an ideal of that ring (because it does not need to be closed under addition).
Cubic graphs are usually defined as finite simple 3-regular graphs. However, the Hamming graph $H(3, 2)$ is also referred by some authors as «The Cubic Graph». Well, it is indeed finite, simple and 3-regular, but not the only one with this property.
The definition of simple graph I am used to is "graph without loops and multiple edges", however I know, that some people define simple graph as "graph without multiple edges" (loops are allowed).
In different sources $D_{2n}$ means either $C_{2n} \rtimes C_2$ or $C_n \rtimes C_2$.
The word "automaton" is sometimes used as a synonym for "acceptor" and sometimes for "transducer"
Sometimes $C_n^k$ means binomial coefficient $\frac{n}{(n - k)!k!}$ and sometimes direct product of $k$ isomorphic copies of a cyclic group of order $n$.
A: Another kind of answer.  There is

increasing; strictly increasing  

and there is

nondecreasing; increasing  

A: 
Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.
  — F. Klein

A: Non-cooperative game theory has the odd property that essentially all authors have their own notions of what a game in extensive form is and thus prove results about principally different mathematical objects.The used notions tend to be equivalent under some reasonable isomorphism, but (almost) nobody ever bothers to make these isomorphisms explicit.
A: Differential geometry seems to have many cases of conflicting conventions:
1) Is the Laplacian a non-negative operator? Or a non-positive one?
2) Is the mean curvature the... mean... of the principal curvatures? Or their sum?
3) Is the directional derivative linear in the magnitude of the direction? Or invariant to it?
A: The notion holomorph in finite group theory
Robert L. Griess writes in [1], Definition 2.21:
A  holomorph of a group $G$ is a group $E$ containing $G$ as a normal subgroup such that $C_E(G) = C(G)$ and the action of $E$ by conjugation on $G$ induces $\mbox{Aut}(G)$.
Later in [1] he explains:
Our term holomorph replaces an older and rare usage which means the semidirect product of a group with its automorphism group.
The term holomorph in the new sense is also used in [2]. The term holomorph in the old sense is used in
https://groupprops.subwiki.org/wiki/Holomorph_of_a_group ,
https://en.wikipedia.org/wiki/Holomorph_(mathematics) .
$ $
[1] Robert L. Griess, Jr., Twelve Sporadic groups, Springer Verlag 1998
[2] A. A. Ivanov, The Monster Group and Majorana Involutions,
    Cambridge University Press 2009
A: Everybody agrees that an isometry is a distance-preserving map. In the context of functional analysis and in particular operator algebras, this is indeed the definition. But in geometry, an isometry is usually required to be bijective, leading e.g. to the isometry group of a metric space.
If there is a danger of confusion, one can still say isometric embedding or isometric isomorphism for disambiguation, whenever no specific term such as unitary is available. (Incidentally, unitary is also an overloaded term meaning either "unitary operator" or "having a unit element".)
A: Positive definite matrix (and related terms).  Most authors require these to be hermitian (or symmetric in the real case), but not all.
EDIT: also, "positive" can be ambiguous even for numbers: most authors, especially in English, consider $0$ to be neither positive nor negative, but according to Bourbaki it is both, and many authors (especially French ones) follow Bourbaki's convention, using "strictly positive" or "strictly negative" for what most authors would call "positive" or "negative".
A: Polygons!
Is a polygon a sequence of vertices together with the edges that connect consecutive vertices?  If so, can two distinct vertices be the same point?  How about two consecutive vertices?  Or a pair of vertices two indices apart: can we go from A, to B, and then directly back to A?
Or if not, can the edges of a polygon intersect each other, aside from the necessary intersections between consecutive edges at vertices?
Can we say that points are inside or outside the polygon?  If so, and the polygon is self intersecting, do the points within the intersection cancel, or stay, or get counted with multiplicity?
Or is a polygon an area whose boundary consists of line segments?  If so, can its boundary be disconnected (can It have holes)?  Can it pinch to a single point like a figure eight?
Are the vertices ordered?  Or at least ordered cyclically?  Are the edges directed?  If so, do the directions of consecutive edges need to match up?
For that matter, do the edges need to connect up at all, or can our polygon have a starting point and an ending point with no edge connecting them?
And that's not even getting into polyhedra...
A: "Function", prior to about 1910 always meant the $y$ in $y=f(x)$ (Look up any definition form that period). Since roughly 1920 it's officially the $f$. Physicists, engineers and many applied mathematicians still mean $y$ when they talk about functions today. But, since the hijacking of the term, they lack adequate terminology to make the distinction between $y$ and $f$. (Probably they are not even aware of the distinction, due to the sloppy use of the word function in most calculus textbooks combined with the common notation $y=y(x)$.)
More on the history of this strange development can be found here and here.
A: Injective: Semigroups can be completely (right/left) injective, while a Banach algebra is injective if the multiplication induces a continuous linear map of the injective tensor product $X\check{\otimes}X$ into $X$ (Varopoulos).  I discovered the former when a reviewer insisted on a hyphen in the title of a paper I had submitted: "Injective seimgroup algebras".
A: An affine stratification can be a stratification into copies of affine space ($\mathbb{A}^{n_i}$) or into affine spaces ($\operatorname{Spec} R_i$).
A: For a commutative ring with unity $R$, a primitive polynomial in $R[x]$ is a polynomial whose coefficients generate the ideal $(1)$. If $R$ is a UFD, a primitive polynomial in $R[x]$ is a polynomial whose coefficients have the greatest common divisor $1$. These defintions coincide iff $R$ is a PID, hence for higher-dimensional UFD's (like $k[x,y]$) we get two different definitions of the same notion.
A perfect ring is a ring, such that every left module admits a project cover. In positive characteristics, a perfect ring is a ring on which the $p$-th power map is an isomorphism.
Also note the slightly different definitions of a projective morphism in EGA and Hartshorne. To be fair Hartshorne adresses the difference after his definition.
Another (not perfectly fitting) example would be a noetherian scheme and a scheme whose underlying topological space is noetherian.
A: In graph theory, there are at least two different meanings of the word "hereditary".
Some definitions first.
Let G be a graph. 
If graph H is obtained by deleting 0 or more vertices from G, then H is an induced subgraph of G.
If graph K is obtained by deleting 0 or more vertices, and deleting 0 or more edges from  G, then H is a subgraph of G. So every induced subgraph is a subgraph, but not vice versa.
Let X be a set of graphs. 
If, for any G in X, any subgraph H of G is also in X, then X is hereditary.
If, for any G in X, any induced subgraph H of G is also in X, then X is induced hereditary.
So every hereditary set is also induced hereditary.
Now, some authors use the term "hereditary" to refer to induced hereditary sets.
A: I believe that the notion of Euclidean domain provides another example of this situation.
Some authors define Euclidean domain as an integral domain $D$ endowed with a function $d \colon D\setminus \{0\} \to \mathbb{Z}^{+}$ such that
(i) If $a,b \in D \setminus \{0\}$ and $a \mid b$, then $d(a) \leq d(b)$.
(ii) If $a, b \in D\setminus \{0\}$, then there exist $q, r\in D$ such that $a=bq+r$ where $r=0$ or $d(r)<d(b)$.
(cf. Herstein's Topics in Algebra, Stewart's Algebraic number theory and Fermat's Last Theorem, etc.). Since this is the definition I was more accustomed to, I must confess I felt a wee bit uneasy when I learnt that, in their book on problems in algebraic number theory, R. Murty and J. Esmonde had chosen to define this concept as follows:

If $R$ is [an integral] domain with a map $\phi \colon R \to
\mathbb{N}$, and given $a, b\in R$, there exist $q, r \in R$ such that
  $a=bq+r$ with $r=0$ or $\phi(r)<\phi(b)$, we call $R$ an Euclidean
  domain.

Later on, I would come to the conclusion that they'd decided to disregard the first condition in the definition à la Herstein because in their examples, etc., they were to analize "euclidianness" with respect to the norm in the extension which determines an application that is even completely multiplicative.
A: Tensor: for some people, a tensor is an element of a tensor product of vector spaces. For others it’s a section of a tensor product of tangent and cotangent bundles on a manifold. Members of the first group would call that latter notion a tensor field. 
More interestingly:
Tensor rank: a tensor of rank $r$ is either a sum of $r$ simple tensors (outer products) or an element of a tensor product of $r$ vector spaces (an “$r$-dimensional array of numbers”). For example, a matrix is a rank $2$ tensor in the latter sense. 
I speculate that at least 25% of Math Stack Exchange questions about tensors are about confusion over one or both of the above. 
A: How about "algebra"? Usually an algebra over a field is assumed to be associative by default, but sometimes it is not. 
Not to mention the various category-theory uses of "algebra" (over a monad, over an operad, for a Lawvere theory, for an endofunctor...).
A: The Fourier transform is defined in at least 3 different ways depending on which subject and school one comes from:
$$
\hat{f}(\xi)=\int_{\mathbb{R}^n} f(x)e^{-2\pi ix\cdot\xi}dx
$$
or
$$
\hat{f}(\xi) = \int_{\mathbb{R}^n} f(x)e^{-ix\cdot\xi}dx
$$
or 
$$
\hat{f}(\xi) = \frac{1}{(2\pi)^{\frac{n}{2}}}\int_{\mathbb{R}^n} f(x)e^{-ix\cdot\xi}dx
$$
The second definition is no longer an unitary operator on $L^2(\mathbb{R}^n)$ and loses the usual symmetry with the inverse, both of which are however rectified again by the third definition.
A: What is a "set"?  ZFC has one answer; ETCS has another; Bishop had another; HoTT has yet another.
A: In the field of dynamical systems, you would see various definitions for a stable system. There are even different definitions for specific names, e.g. Lyapunov stability in the literature.
I have seen that some call the limit cycle a stable manifold, while others consider it an unstable one.
A: To many authors a "category" is necessarily locally small, but to others it need not be.
A: Some authors formulate separability axioms T3/T4 as normality/regularity plus T1. Some other authors do not require T1, on the contrast, they define normality/regularity as T3/T4 plus T1. 
A: "Topos" sometimes means "elementary topos" and sometimes "Grothendieck topos".
A: Simply connected : for some authors, such a space is necessarily path-connected, for others, not.  
A: Singular support of a sheaf seems to be a subset of the cotangent bundle, whereas the singular support of a distribution is a subset of the base space. The former is more like the wavefront set, as far as I can intuit.
People try to avoid the confusion by denoting the former by S.S., and the latter by sing supp, but in my mind they read the same.
A: For some people, the dihedral group $Dn$ has order $n$ and for others, its  order is 2$n$.  See the discussion on MathStackExchange.  I'll look it up when I get a chance.
A: A partition of a set.
A partition of an integer.
A partition of an interval (in defining Riemann and Riemann–Stieltjes integrals.
A partition of unity.
A partition of a sum of squares in multiple regression, multi-way analysis of variance, experiments with both random effects and fixed effects, the partition of residual sums of squares into pure-error and lack-of-fit sums of squares, etc.
A partition of a matrix.
Quotition versus partition.


*

*$6\div2=3$ because $3$ is how many $2$s must be added to get $6$ (quotition).

*$6\div2=3$ because when $6$ is a sum of $2$ equal terms, then each term is $3$ (partition).

A: A tree can be a very different thing in different parts of mathematics. It might be a certain kind of acyclic graph; or a partial order such that the predecessors of every node are linearly ordered; or a partial order where the predecessors are well-ordered; or either of these, except with successors instead of predecessors. In some parts of mathematics, trees are presumed finite, while in others, they are nontrivial only when infinite.
A: Topology: Some authors use the term neighborhood while other use open neighborhood instead.
A: I would say the word "kernel" is probably among the most overloaded terms in mathematics. You've got kernels of linear operators, convolutional kernels, distribution kernels, Markov kernels, and Reproducing Hilbert Space kernels. All of these notions are related but are, strictly speaking, distinct objects. Thus, the kernel of a linear operator is a subspace, while an RKHS kernel is a positive definite map from some set times itself to the reals.
A: $(a,b)$
Is that a coordinate pair representing a point in the plane? or, 
The open interval from $a$ to $b$? or 
The greatest (highest) common factor (divisor) of $a$ and $b$? or 
The ideal generated by $a$ and $b$? or
The inner product of $a$ and $b$? or...
A: Antisymmetrization and symmetrization of tensors. Should we divide it by $(n!)$ ? This affects the relations between tensor and (anti-)symmetric algebra, the theory over $\mathbb Q$ and $\mathbb Z$ and is generally a mess. A special case: is a quadratic form over $\mathbb Z$ represented by a polynomial with integer coefficients or by a symmetric matrix with integer elements?
A: Two examples from point-set topology:


*

*a topological space $(X,\tau)$ is locally compact if "any point has a compact neighborhood" or if "any point has a local basis of compact neighborhoods". The two are equivalent if the space is Hausdorff, but not in general!

*Even more subtle: a topological space $(X,\tau)$ is locally connected at $x$ if "$x$ has a local basis of connected neighborhoods" or if "$x$ has a local basis of open connected neighborhoods". The two are equivalent if we ask them for all points $x\in X$, but not in general!

A: Perhaps a prominent example is the definition of a smooth manifold. Some authors require the underlying locally Euclidean Hausdorff space to be 2nd countable, while others require it to be only paracompact. The latter is more general since it allows for uncountable number of connected components, although I'm not sure if that makes such a big difference in the everyday life of a differential geometer. But usually there is no room for confusion since most authors state at the beginning which convention they use.
EDIT: It turns out it actually does make a pretty big difference, see Stefan Waldmann's comment below.
A: For some strange reason, algebraic order theory choose "lattice theory" as its name, and "lattice" as its central object of study. But of course, for everybody else "lattice" still means what it always meant, and what it also means in plain English.
A: The definition of a Turing Machine is a great example, where multiplied all together there are at least hundreds of possible definitions.


*

*Is the tape doubly infinite or singly infinite? (If singly infinite, what happens when you move right at the right at the end of the tape?)

*Does the machine have special "accept" and "reject" states, just a "halt" state, or neither of these?

*Is the input string written on a separate read-only tape? Is the output string written on a separate write-only tape?

*Does the number of states include the accept and reject states?

*Does the space usage include input and output? Does the running time include the step when the machine halts?

*Is the input alphabet $\{0,1\}$, or an arbitrary set? May it contain only 1 symbol or even 0 symbols?

*Is the tape alphabet just adding a blank symbol, or may it contain other convenient working symbols?

*Are multiple tapes allowed? Is staying still instead of moving left or right allowed?
One usually spends a week in a theory of computation course fleshing out some of the equivalences between various definitions. But I don't know that anyone has ever done this in machine-checked or at least formally exhaustive detail, avoiding all pitfalls and edge cases. Rather, researchers rely heavily on the informal Church-Turing thesis and assume some kind of programming language or machine representation which suits their needs and is equivalent to all these definitions of Turing Machine at once.
Yet they are not at all equivalent for many purposes. (1) If worried about low-level running time (not just up-to-a-polynomial), doubly-infinite vs singly-infinite, how many symbols are allowed, multiple vs. single tapes, and whether accept/reject is part of the finite state machine all have an impact. (2) If worried about sub-linear complexity classes, we know that it is crucial that the input string be read-only. (3) Even if only worried about defining polynomial time computability, at least two symbols are required in the input alphabet. (4) To define the Busy Beaver function, as well as many other non-computable functions, requires a specific representation to be fixed. This one is a big "gotcha" that one always runs into when asking students (as an exercise) to compute the Busy Beaver function of $1$ or $2$.
A: The notion of spectrum in operator theory. Some people assume that if $\lambda$ has a property that $T-\lambda I$ is injective with dense range which is not the whole space and with continuous inverse then $\lambda$ is not an element in spectrum since $(T-\lambda I)^{-1}$ can be extended continuously other authors consider such $\lambda$ as an element of spectrum. 
A: I'm surprised nobody has mentioned "range" which can by synonymous with "co-domain" or "image" depending on personal preference.
A: In set theory a forcing notion can be a pre order with a largest element or lowest element, depending on the style of the author. 
A: Speaking of functional analysis: While algebraists seem (at least as far as I can tell) to finally accept that things called ring and in particular things called algebra really should have units, people in functional analysis seem adamantly opposed to this, especially those who work with $C^\ast$-"algebras". I can sort of see why they do it, but I don't find their reasons particularly compelling; they rarely seem to talk about actual honest to god $C^\ast$-algebras-without-unit. Most of the time an "algebra" is really better viewed as an ideal in some proper algebra. All the proofs I have seen work the same way: First step is always to go to the unital case by embedding the "algebra" in question into an unital algebra.
As an algebraist myself, talking to a friend of mine who lives in $C^\ast$-land sometimes is a hassle because of this.
A: In a real vector space $E$, some people use cone to mean a subset $C \subseteq E$ closed under multiplication by positive reals. If it is additionally closed under addition, they call it a convex cone, and if $C \cap -C = \{0\}$, they say it is a proper cone, or that $C$ is proper. 
A second group of people has no name for the first thing, but calls the second thing just a cone, also using proper cone for the third thing.
A third group of people also never use the first thing, but call the second thing a wedge and the third thing a cone. So instead of saying "we prove that this cone is proper", they say "we prove that this wedge is a cone". 
Since the preorder defined by a convex cone is a partial order iff that cone is proper, I am inclined to put myself in the third group of people and use the short word cone for the most useful notion.
A: The term 'Macdonald polynomial' might refer to:


*

*The symmetric function $P_\lambda(x;q,t)$ with coefficients being rational in $q,t$. 

*The symmetric function $J_\lambda(x;q,t)$, being $P_\lambda(x;q,t)$ multiplied by a normalization factor.

*The one of the non-symmetric polynomial $E_\alpha(x;q,t)$ (Haglund vs. Marshall normalization/convention).

*The non-symmetric polynomial $E_\lambda(x;q,t)$ indexed by weights in a root lattice, and there is a formula over alcove walks.

*The modified Macdonald polynomial $\tilde{H}_\lambda(x;q,t)$, which is the most 'combinatorial' version, which is the bigraded Frobenius character of a certain $S_n$-modules indexed by $\lambda$. Combinatorial formula due to Haglund.

*There are also non-homogeneous Macdonald polynomials...
This specializes to confusion regarding Hall-Littlewood polynomials (the 'standard' $P_\lambda$, the 'dual' $Q'_\lambda$ or the modified, $\tilde{H}_\lambda$).
A: An extension of a group $A$ by a group $B$ can be either a group $G$ with a normal subgroup isomorphic to $B$ with $G/B$ isomorphic to $A$ or a group $G$ with a normal subgroup isomorphic to $A$ with $G/A$ isomorphic to $B$.  (See, for example, https://en.wikipedia.org/wiki/Group_extension especially section 1.2.3.)  
A: Perhaps the mother of all examples is "natural number". You can start an internet flame war by asking whether zero is a natural number. 
A: The limit of a function may be a deleted limit or a non-deleted limit.

A number $L$ is the deleted limit of $f$ as $x$ approaches $p$ if, for every $\epsilon>0$, there exists a $\delta>0$ such that $0<|x−p|<\delta$ and $x$ is in the domain of $f$ implies $|f(x)−L|<\epsilon$.

The non-deleted limit is exactly the same except that instead of $0<|x-p|<\delta$ you have $|x−p|<\delta$.
For example, if
$f(x)=\begin{cases} 1,&x=0\\0,&x\ne 0\end{cases}$,
then $\lim_{x\to 0}f(x)=0$ if deleted limits are used, but $\lim_{x\to 0}f(x)$ does not exist if non-deleted limits are used.
A: Linear functions:
In high-school algebra (sometimes called "pre-calculus"), we are taught that linear functions are those of the form $y=mx+b$, because they are graphed by a straight line in the plane.
Then we study linear algebra in university, and realize that for a function $f$ to be linear it must satisfy $f(0)=0$, as a  special case $a=0$ of the linearity requirement that $f(ax)=af(x)$, and the functions that were called "linear" in high school are not really linear (unless $b=0$) but affine.
A: Variety has a number of slightly different definitions. Apparently some authors use reduced of finite type over a field, whereas I would want separated to rule out the line with two origins. Most people use scheme-theoretic language to solve this problem.
Conway-Sloane jokes the discriminant of a quadratic form is a function of the form and the author. They are not wrong. That same subject has two distinct definitions of integral, which are basically the same (multiply by $2$) but which are a pain to keep track of. The Hasse-Witt invariant has two different definitions as well.
A: Some people take division ring, division algebra, and skew field to be the same thing; namely a (possibly noncommutative) ring with $1\neq 0$ such that all nonzero elements are units.  Some authors however add one or two extra qualifiers to some of these definitions.  [Such as forcing noncommutativity, or being finite dimensional over the center, or...]
A: The distinction between compact and quasi-compact spaces is another one. If a topological space has the property that any open cover contains a finite subcover, then to some people this space is compact, to others it is quasi-compact. In the latter case, a compact space would then be a quasi-compact space that is also Hausdorff.
A: I believe the number of proper definitions of the term "quantum group" is some element of $\mathbb R \setminus \{1\}$. Even the simplest example, the quantized enveloping algebra of $\mathfrak{sl}_2(\mathbb C)$, has at least two definitions (more if you count the coproduct as part of the definition). 
If you are careful with the details, as for example in Jantzen's book "Introduction to Quantum Groups", you either get bogged down in the details of left-right compatibilites and twisted coproducts or, more in the spirit of the subject, you change the definition midway.
A: Are Hermite polynomials $H_n(x)$ orthogonal w.r.t. the weight $e^{-x^2}$ or $e^{-x^2/2}$?
A: What some call the adjugate matrix is often called the adjoint (or sometimes classical adjoint) matrix, while adjoint matrix commonly also means the transpose or Hermitian transpose. I doubt this double use ever causes confusion to anyone but students, but it is an example of the sort the questioner seeks.
A more serious example is that regular elements of (algebraic) Lie groups are defined differently by different authors. Correspondingly one encounters various definitions of what is a Cartan subalgebra. Here the issue tends to be the scope of definition; usually two different definitions are equivalent in the semisimple setting, or over characteristic zero, but not in more general settings.
A: I can't believe this hasn't mentioned already.
What is a graph?
Everybody agrees that a graph has vertices and edges (which are usually not oriented). Depending on whom you ask, it might not be allowed to have multiple parallel edges between two vertices. Often a graph is not allowed to have loops at a vertex. Also, is a graph necessarily finite?
And I am not even talking about all the non combinatorial definitions of a graph, e.g. $1$-dimensional CW-complex.
A: Not a word but a piece of notation: Sometimes I have seen $\subset$ used to mean "is a proper subset of" while other times I have seen it used to mean "is a subset of".
A: The notion of a category: 
It sometimes happens that some people define a category as a locally small category without stating it, or probably without knowing that the general definition is where a category consists of a class of morphisms, instead of that morphisms between any two of its objects has to be a set. Thus, if you see the term "category" in the literature/ in lecture notes, the definition of a category might denote a category which is locally small.
The notion of a topological category is another example from this area https://en.wikipedia.org/wiki/Topological_category.
Furthermore, it often happens that there are definitions of something which are equivalent for cartain classes of objects, but not in general:
In $C^*$-algebras: There are notions like "stably finite" (and other notions like this), which can be non-equivalent, see here https://math.stackexchange.com/questions/2073741/stably-finite-c-ast-algebras for instance, which are defined in many ways and which are  equivalent for a certain classes of $C^*$-algebras (simple $C^*$-algebras in this case).
The notion 'quasidiagonality' of $C^*$-algebras is another example.
A: A distribution can either be related to probability or to generalized functions
A: In functional analysis, an embedding of one topological vector space $X$ into another $Y$ is simply a continuous linear injection $T$ from $X$ to $Y$.  In particular it typically is not an embedding in the usual sense of general topology, since $T$ need not be a homeomorphism onto its image.
A: When is a function concave? When is it convex? Do you determine this by looking at the graph "from above", or "from below"?
A: The wide range of choices in the definition of an automorphic form is particularly annoying. Depending on the purposes, it could be a meromorphic function fully invariant by a certain discrete group of transformation, a holomorphic function almost-invariant, a differential form, a subrepresentation of an $L^2$ space, a classical or an adelic object,  a solution to a partial differential equation, etc.
These are sometimes related, sometimes definitely different, and the lack of vocabulary consistency in such an active field requires some care when dealing with object (leading authors to usually restate a precise definition of the objects they are dealing with, or at least quoting the literature associated with the paradigm they have chosen.
A: Let $X_1,X_2,X_3,\ldots$ be independent random variables.
Sometimes it is said that a stopping time for this random process is a random variable $T$ for which the truth value of $T=n$ (for $n=1,2,3,\ldots$) is determined by the values of $X_1,\ldots,X_n.$
And sometimes it is said that a stopping time for this random process is a random variable $T$ for which the each of the events $\big[T=n\big]$ for $n=1,2,3,\ldots$ is independent of the sequence $X_{n+1}, X_{n+2}, X_{n+3},\ldots\,.$
The second definition is more inclusive than the first.
