What is the name for a point that is periodic to within $\varepsilon$? Let $X$ be a set and $f: X \to X$ a function. A point $x \in X$ is, of course, said to be periodic for $f$ if $x \in \{f(x), f^2(x), \ldots\}$.
Now suppose that $X$ is a topological space and $f$ is continuous. In private, I've been saying that a point $x \in X$ is topologically periodic for $f$ if $x \in \overline{\{f(x), f^2(x), \ldots\}}$.
But "topologically periodic" is just a name I made up.

What is the standard name for "topologically periodic" point?

Note that I'm looking for an adjective that applies to points, not functions. I'm aware of terms such as quasiperiodic function and almost periodic function, but they don't answer my question.
In case it helps, I'm working in less generality than arbitrary topological spaces and continuous maps. In my situation, $X$ is a compact metric space and $f$ is distance-decreasing ($d(f(x), f(y)) \leq d(x, y)$). Under those hypotheses, my definition of "topologically periodic" may have other equivalent formulations that are more familiar.
 A: TL;DR: There is no standard name in the topological dynamics literature for the property in question; however currently "recurrent point" seems to be the most common.
As Ville Salo wrote above, a point returning once is equivalent to a point returning infinitely many times, in the context of continuous selfmaps of compact metric spaces and much more.
Below I list some references and point out to a certain related discrepancy in the topological dynamics literature regarding terminology. I also add a personal justification as to why one would prefer the syntactically more complicated definition.
My answer ended up being quite long; I hope the length will not be considered as impolite. As can be seen above I initially made a mistake in the comments; and my answer (or its length) should not be taken as me not owning up to my mistake.

First let me give some definitions; I'll work in a more general framework (although not the most general).
Let $T$ be a pointed topological space, $X$ be a topological space, $\mathcal{F}_\bullet\subseteq F(T;F(X;X))$ be a family of (not necessarily continuous) selfmaps of X parameterized by $T$ (not necessarily continuously) with the property that $\mathcal{F}_t=\operatorname{id}_X $ iff $t=e_T$. For $t_\bullet\subseteq T$ a net, say that $t_\bullet$ goes to infinity if $t_\bullet$ has no limit points. Suppose there is at least one $t_\bullet$ that goes to infinity. (Thus $T$ models infinite time, $X$ models space, $\mathcal{F}$ models time evolution.)
Put $T^\ast=T\setminus\{e_T\}$ and for any $A,B\subseteq X$ define the set of hitting times (or the return set, or the dwelling set, or the meeting time set) from $A$ to $B$ by
$$\operatorname{Hit}^\mathcal{F}(B\leftarrow A)=\{t\in T^\ast\,|\, \mathcal{F}_t(A)\cap B\neq \emptyset\}.$$
For $x\in X$ we also define $\operatorname{Hit}^\mathcal{F}(B\leftarrow x)= \operatorname{Hit}^\mathcal{F}(B\leftarrow \{x\})$.
The qualitative ($\dagger$) notions of recurrence in question revolve around the hitting sets $\operatorname{Hit}^\mathcal{F}(N\leftarrow x)$ for $N$ an arbitrary neighborhood of $x$. Indeed consider the following three properties for $x\in X$ an anonymous point:

*

*$\forall N\in\operatorname{Nbhd}(x): \operatorname{Hit}^\mathcal{F}(N\leftarrow x)\neq\emptyset$.

*$\forall N\in\operatorname{Nbhd}(x): \operatorname{Hit}^\mathcal{F}(N\leftarrow x)$ contains a net going to infinity.

*$\forall N\in\operatorname{Nbhd}(x),\exists$ compact $K_N\subseteq T:$ $T= K_N \cdot\operatorname{Hit}^\mathcal{F}(N\leftarrow x)$.

(The first property models one return, the second property models infinitely many returns, and the third property models infinitely many returns with bounded gaps in time.)
(Apart from its day-to-day utility, one could argue that using hitting sets to formulate things is preferable due to categorical reasons, at least in the case when the dynamics is autonomous; see e.g. Ex.5 at Tao's blog https://terrytao.wordpress.com/2008/01/10/254a-lecture-2-three-categories-of-dynamical-systems/ along these lines.)

Let us now point out the classical cases; for this paragraph $T$ will be a monoid or group, $\mathcal{F}_\bullet$ will be a monoid or group homomorphism (function composition will be the operation in the target) that unCurries to a family in $C^0(T\times X;X)$.
$T=\mathbb{Z}_{\geq0},\mathcal{F}_\bullet=f^\bullet$:

*

*$\iff x\in \overline{\mathcal{O}_{f(x)}(f)}\, \,\,$ ($=$ closure of the orbit of $f(x)$). (This is the property Prof. Leinster was asking about.)


*$\iff x\in \bigcap_{n\in\mathbb{Z}_{\geq0}} \overline{\mathcal{O}_{f^n(x)}(f)} \stackrel{\tiny\text{def}}{=} \omega_x(f)$.


*$\iff \forall N\in\operatorname{Nbhd}(x),$ $\exists l=l_{N}\in\mathbb{Z}_{\geq1},$ $\forall n\in\mathbb{Z}_{\geq0},$ $\exists i=i_{N,n}\in \{0,1,...,l-1\}:$ $f^{i+n}(x)\in N$.
$T=\mathbb{R}_{\geq0},\mathcal{F}_\bullet=\varphi_\bullet$:

*

*This property holds for any point by continuity; there are multiple ways to model "one return" (hence implicitly "one departure"), but all such properties seem to  necessarily refer to an anonymous neighborhood of the point.


*$\iff x\in \bigcap_{t\in\mathbb{R}_{\geq0}} \overline{\mathcal{O}_{\varphi_t(x)}(\varphi)} \stackrel{\tiny\text{def}}{=} \omega_x(\varphi)$.


*$\iff \forall N\in\operatorname{Nbhd}(x),$ $\exists l=l_{N}\in\mathbb{R}_{\geq0},$ $\forall t\in\mathbb{R}_{\geq0},$ $\exists s=s_{N,t}\in [0,l]:$ $\varphi_{s+t}(x)\in N$.
Finally for the cases of $T=\mathbb{Z},\mathcal{F}_\bullet=f^\bullet$ and $T=\mathbb{R},\mathcal{F}_\bullet=\varphi_\bullet$, the third properties are analogously paraphrased. For the first and second properties one makes use of the two infinities in time (modelling "before time begins" and "after time ends"), and defines also $\alpha$-limit sets using negative time, and uses the adverbs "positively" and "negatively" appropriately. (In the definition of limit sets the intersections can be taken over the whole group.)
(As a note for actions of more general groups, I would presume one would refer to some monoid cover of the group that is derived more or less canonically from certain extra properties of the action; think e.g. Weyl chambers; the main point is that in general algebraic/geometric/topological infinities in time and dynamical infinities in time may fail to coincide. There are indeed papers along these lines e.g. from the Gottschalk-Hedlund era; also see the paper by Auslander, Glasner & Weiss mentioned below.)

In this paragraph I give a slightly shorter proof that in quite a general context one return is equivalent to infinite returns. In the comments above I had alluded to the necessity of some separation axiom, but for this equivalence it's not needed (there are other cases where one needs to turn one return to infinite return where some separation is necessary; roughly speaking some separation axiom would be needed if one needs to control the return times, and consequently departure times; thus heuristically we don't need any separation axiom because the point doesn't need to go elsewhere).
Claim: Let $X$ be a topological space, $f:X\to X$ be a function. If $f$ is continuous, then for any $x\in X$:
$$x\in \overline{\mathcal{O}_{f(x)}(f)} \Longleftrightarrow x\in \omega_x(f).$$
(Consequently for topological actions of $\mathbb{Z}_{\geq0}$, the first two properties above are equivalent.)
Proof: $(\implies)$ is syntactic. For $(\impliedby)$, suppose $x\in \overline{\mathcal{O}_{f(x)}(f)}$, and let $U$ be an open neighborhood of $x$. Then for some $k_1\geq 1$, $f^{k_1}(x)\in U$. By the continuity of $f$, $f^{-k_1}(U)$ is also an open neighborhood of $x$. By induction we have a sequence $k_1,k_2,...$ of positive integers such that for $K_n=k_1+k_2+\cdots+k_n\geq n$ with $n\geq 1$, $f^{K_n}(x)\in U$. Then we have that $f^{K_n}(x)\in\mathcal{O}_{f^n(x)}(f)\cap U$.

As two remarks, with some separation one obtains the equivalence of one return and infinite returns, even when the dynamics is not continuous:
Claim: Let $X$ be a topological space, $f:X\to X$ be a function. If $X$ is $T_1$ (= given any two distinct points, there is a neighborhood of the first that doesn't contain the second), then for any $x\in X$:
$$x\in \overline{\mathcal{O}_{f(x)}(f)} \Longleftrightarrow x\in \omega_x(f).$$
Proof: If $x\in \overline{\mathcal{O}_{f(x)}(f)}\setminus \omega_x(f)$, then for any $N\in\operatorname{Nbhd}(x)$, there is an $n_N\geq 1$ such that $f^{n_N}(x)\in N$, and there is some $n^\ast\geq2$ and $N^\ast\in\operatorname{Nbhd}(x)$ such that for any $n\geq n^\ast: f^n(x)\not\in N^\ast$. Thus
$$\forall N\in\operatorname{Nbhd}(x): \{f^{1}(x),f^{2}(x),...,f^{n^\ast-1}(x)\}\cap N\cap N^\ast\neq\emptyset.$$
Since $X$ is $T_1$, we have that $x\in \{f^{1}(x),f^{2}(x),...,f^{n^\ast-1}(x)\}$, so that $x$ is periodic; a contradiction.
Something needs to be assumed though:
Example: Put $X=\mathbb{Z}_{\geq0}$, $f:X\to X, x\mapsto x+1$. Topologize $X$ via $\{\,\,\{0,1\},\,\, \{1\},\,\,\{2\},\,\,\cdots\,\, \{n\},\,\,\cdots\,\,\}$. Then $X$ is not $T_1$ (though it is $T_0$ (= given any two distinct points, there is a neighborhood of one of them that doesn't contain the other one)), $f$ is not continuous (though it is Borel measurable), and
$$\overline{\mathcal{O}_{f(0)}(f)}=X;\,\,\,\omega_0(f)=\emptyset.$$
In particular, $0$ satisfies Property 1 but not Property 2.
(I was playing with variants of this example before I realized continuity would be sufficient for equivalence; the effect similar to the so-called "chain fountain" (see https://en.wikipedia.org/wiki/Chain_fountain); continuity makes the whole orbit fall into the topologically pathological part.)

On the other hand, even in reasonable situations Property 3 seems to be strictly stronger. E.g. consider the expanding map $f:x\mapsto 2x$ on $\mathbb{T}$. The orbits under $f$ of some but not all points are dense (i.e. $f$ is "topologically transitive" but not "minimal"). Any point with dense orbit satisfies the second (hence also the first) property, but no such point satisfies the third property. This is roughly because Property 3 implies that one can cover the closure of the whole orbit by finitely many preimages of a closed neighborhood. See the Birkhoff Recurrence Theorem (Thm.4.2.2 on p.170) in the book by de Vries mentioned below for details; in general for a non-minimal but topologically transitive continuous selfmap of a $T_3$ ( = Hausdorff and points & closeds can be separated) space any transitive point will always satisfy the first two properties but never the third property (see Ex.2 on p.171 of the same book).
However it is also not the case that all three properties are never equivalent; see e.g. Auslander, Glasner & Weiss' "On recurrence in zero dimensional flows".

Let me now list some references regarding what names people give to the properties above, disregarding the situations when these properties may be equivalent. The list is certainly not comprehensive.
Let $f:X\to X$ be a continuous selfmap of a topological space, $x\in X$.
$x$ is recurrent if it satisfies Property 1 above, according to:

*

*Brown's Ergodic Theory and Topological Dynamics, p.47,


*Furstenberg's Recurrence in Ergodic Theory and Combinatorial Number Theory, p.20,


*de Vries' Topological Dynamical Systems, p.165. (Interestingly, in his earlier book Elements of Topological Dynamics, de Vries calls a point recurrent if it satisfies a property that is formally more similar to Property 2 ("some point in the orbit of $x$ satisfies Property 2"); and after mentioning equivalence to Property 2 he declares that Property 2 "explains the nomenclature", see pp.57-58.)


*Nitecki's Differentiable Dynamics, p.22.
$x$ is recurrent if it satisfies Property 2 above, according to:

*

*Gottschalk & Hedlund's Topological Dynamics, p.26 (it might be useful to look at their paper "The Dynamics of Transformation Groups", p.349 for a less optimized definition)


*Katok & Hasselblatt's Introduction to the Modern Theory of Dynamical Systems, p.129,


*Brin & Stuck's Introduction to Dynamical Systems, p.29,


*Walters' An Introduction To Ergodic Theory, p.157,


*Einsiedler & Ward's Ergodic Theory: with a view towards Number Theory, p.104


*Milnor's notes at https://www.math.stonybrook.edu/~jack/DYNOTES/dn4a.pdf, p.4-4.


*Tao's blog at https://terrytao.wordpress.com/2008/01/08/254a-lecture-1-overview/ (see Birkhoff Recurrence Theorem there; this definition (of "recurrence", not of "Birkhoff Recurrence Theorem") recurs throughout the same series of blogged lecture notes.)
$x$ is recurrent if it satisfies Property 3 above, according to:

*

*Birkhoff's Dynamical Systems, p.199

*Nemytskii & Stepanov's Qualitative Theory of Differential Equations, p.375 (at least when $X$ is compact metric)

*Sell's Topological Dynamics and Ordinary Differential Equations, p.90.

*Koo's 2019 paper "Recurrence and Stability of Points in Discrete Flows", p.252 (as a token of contemporary use).

On the other hand, $x$ is Poisson stable if it satisfies Property 2 above, according to

*

*Birkhoff's book, p.190 (strictly speaking this is "a kind of Poisson stability"; true Poisson stability for Birkhoff is the conclusion of Poincaré Recurrence; infinite returns a.e., see p.174)

*Nemytskii & Stepanov's book, p.22,

*Sell's book, p.85.

Above I have suppressed some equivalences and special/general cases; de Vries' book contains further historical information as well as a more extensive Rosetta Stone on pp.214-215. Likewise there are some comments on terminology in Sell's book on p.92 as well as Furstenberg's book on p.15.

Let me now make a few comments as to why it seems to me Property 2 is more deserving of the name "recurrence". One of the reasons why this answer got this long is that I got curious as to why some mathematicians were using the Property 2 instead of the simpler Property 1 for recurrence. The following three reasons are the ones I could come up with; they are all eventually personal of course, but each seem to have different slants:

*

*(Practical) Property 2 is more useful in practice; especially if one wants to stack up a few properties often one needs infinite returns.


*(Mathematical) To flows and beyond, as well as to multiple/simultaneous recurrence, Property 2 generalizes easier. Similarly Property 2 is easier to modify to come up with quantitative versions (see below).


*(Philosophical) Poincaré Recurrence divides those who are aware of it into two: for one group it refers to the fact that $\{x\in A\,|\, \operatorname{Hit}^f(A\leftarrow x)\neq\emptyset\}=_\mu A$, and for the other group it refers to the fact that $\{x\in A\,|\, \operatorname{Hit}^f(A\leftarrow x)=\infty\}=_\mu A$. I seem to be a champion of the latter group. If I remember correctly Ghys makes some comments along these lines in the beginning of his talk on Poincaré Recurrence on Youtube (as far as I remember it's one of his favorite theorems).

($\dagger$) Finally let me add a paragraph about quantitative notions of recurrence. Indeed, what I've written above is quite far from the current paradigm.
First note that if $X$ were metrizable, then Property 2 would be equivalent to
$$\sup_d \liminf_{n\to \infty} d(f^n(x),x)=0,$$
where the first supremum is over the distance functions $d$ compatible with the topology of $X$.
This gives one way of quantification: quantification in terms of proximity (e.g. one can and does consider the decay of distances against a weighting function). One can also introduce a measure to quantify in terms of time. Of course one ought to expect these two ways of quantification to be closely related. There are very interesting papers along these lines, e.g. Boshernitzan proved in "Quantitative recurrence results" (p.618) that if $X$ is metrizable, $d$ is a compatible distance function on $X$, $f:X\to X$ is a Borel measurable self-map of $X$, and $\mu$ is a Borel probability measure on $X$ invariant under $f$, then
$$\forall \alpha\in \mathbb{R}_{>0}: \operatorname{Haus}_\alpha^d(X)=0\implies \forall x\in_\mu X: \liminf_{n\to \infty} n^{1/\alpha}\, d(f^n(x),x)=0,$$
where $\operatorname{Haus}_\alpha^d(X)$ is the $\alpha$-Hausdorff measure of $X$ w/r/t the distance function $d$.
A: Such a point is called recurrent.

It was stated by Alp Uzman in a comment that the usual definition of recurrent is that $x$ is in its own $\omega$-limit set. This is the definition I had in mind, and under a suitable separation axiom it's obvious that this is equivalent to what was asked, which is why I just wrote the statement. Actually you don't need any separation axiom. I'll just write a full proof.
The $\omega$-limit set $\omega(x, f)$ of $x$ is the set of accumulation points of the sequence $I_x = (x, f(x), f^2(x), \ldots)$. (In the reference given by Uzman, read literally the first definition of the $\omega$-limit set given is that it's the accumulation points of the set $\{f^i(x) \;|\; i \in \mathbb{N}\}$. This would not be equivalent, and is "wrong", but it's clear from the discussion that the author means accumulation points of the sequence.)
We need an auxiliary notion. For two points $x, y$, write $y \leq x$ if every open set containing $x$ also contains $y$. We need two lemmas, which are left as exercise.

Lemma A. Continuous functions preserve $\leq$ in the sense that $y \leq x  \implies f(y) \leq f(x)$.


Lemma B. The $\leq$-relation is transitive.


Theorem. Let $X$ be a topological space and $f : X \to X$ a continuous function. Then $x$ is in $\omega(x, f)$ if and only if $x \in \overline{\{f^i(x) \;|\; i \geq 1\}}$.

Proof. Clearly $\omega(x, f) \subset \overline{\{f^i(x) \;|\; i \geq 1\}}$, so it suffices to show $x \in \overline{\{f^i(x) \;|\; i \geq 1\}} \implies x \in \omega(x, f)$. Suppose thus $x \in \overline{\{f^i(x) \;|\; i \geq 1\}}$.
First, suppose that $f^n(x) \leq x$ for some $n \geq 1$. Then by Lemma A, $f^{(k+1)n}(x) \leq f^{kn}(x)$ for all $k$. By Lemma B, $f^{kn}(x) \leq x$ for all $k$. Then clearly $x$ is even an accumulation point of the subsequence $(x, f^k(x), f^{2k}(x), \ldots)$ -- all points of this set are in every neighborhood of $x$.
Next, suppose $f^n(x) \not\leq x$ for all $n \geq 1$. For all $n \geq 1$ let $V_n$ be a neighborhood of $x$ not containing $f^n(x)$, and $W_n = \bigcap_{i = 1}^n V_i$. Let $U \ni x$ be any neighborhood. By the assumption on $x$, for any $n$ we have $f^m(x) \in U \cap W_n$ for some $m \geq 1$. Necessarily $m > n$, so in particular we find infinitely many $m$ such that $f^m(x) \in U$. Square.
