I confess I really like this question, because it troubled me a lot when trying to teach (and understand) beginning algebraic geometry. There seem to be two questions here: 1) which is the more fundamental notion, morphism or rational map; and 2) what special definition is preferred in the case of quasi projective varieties? I finessed these difficulties for years by working largely in the category of complex analytic varieties and appealing to Chow's theorems, so I am not expert, but I have thought about it. I will share my necessarily naive conclusions, so I can learn from the responses.

As to 1), the notion of rational map seems more fundamental except in the one case of polynomial maps of affine varieties. I.e. if one wants the concept of regularity to be local, one must define it using rational functions. This arises as soon as one defines a regular function on even a quasi affine variety. So it appears as if regular functions are naturally a subclass of rational ones. I.e. first one must understand rational functions, and the set on which such a function is regular.

If the varieties are embedded in projective space as closed sets, one could use the standard affine open cover and stick to polynomials, but if they are abstract or even quasi projective, it seems to require a notion of isomorphism, or of the sheaf of regular functions to define an affine open set.

Note Ruadhai's definition above requires the use of rational functions to define that of regular ones, after which the definition of morphism appears to involve only the concept "regular".

In case 2) we are dealing with projective space, which has two representations, as a space with a natural affine open cover, but also as a space which is a group quotient of an open subset of affine space. Hence there are two natural definitions of regular map, locally regular in terms of the affine cover, but also as a regular map of the big affine space which is constant on orbits. the latter definition yields a map by homogeneous polynomials of the same degree. This definition is the one most convenient for writing down examples.

Again, Ruadhai's definition of regular locally rational function uses the homogeneous representation, so that definition of morphism is an amalgam of abstract sheaf theoretic concepts and concrete projective ones.

The fully concrete version of Ruadhai's definition of morphism is thus: a map locally defined by homogeneous polynomials of the same degree with non vanishing denominators. This is the definition actually used in practice to write down morphisms of projective and quasi projective varieties.

Several basic books give both definitions (locally rational and regular for the affine open cover, and local homogeneous representation) in the quasi projective case, Shafarevich, Harris, and Reid, I believe. This seems to me a helpful practice.

Bill Fulton's lovely little book on curves, which is intended to prepare one for schemes, uses only the abstract open cover approach as definition, but then leaves as exercise to the reader to check that such innocent maps as [x:y]-->[x:y:z] from P^1 to P^2 are regular.

To summarize, it seems one cannot avoid local definitions, and if one wants to avoid fractions altogether, the only definition I know of a morphism of quasi projective varieties is as a map given locally by homogeneous polynomials of the same degree without common zeroes.

In my example above of a map from the intersection of two quadrics to a plane cubic the map [x:y:z:w] to [x:y:z] is undefined at [0:0:0:1]. By using the locally rational regular map definition in affine coordinates, it seems that point goes to [-1:0:1], as given by the hopefully equivalent map [w(y-w) : (x-z)(y-w) : w^2 ].

Edit:This is my attempt to understand Donu’s definition above.
Now I think I understand it, as well as why it is a benchmark, and yet not his favorite. Of course I cannot speak for him. One of its beauties as he said is that it does mimic closely the familiar chart/coordinate system definition from analytic and differential geometry.

Its shortcoming is perhaps its cumbersome nature. E.g. it requires an infinite family of coordinate charts even to discuss regular functions on (open subsets) of affine space. As to whether it enables one to dispense with rational as opposed to polynomial maps, well yes and no. It is true that all regular maps are locally polynomial in terms of the charts, but the trick is that the charts themselves are defined by rational functions.

E.g. Suppose I want to glue two copies of A^1, along the complement of their origins, to get P^1. Can this gluing be done by polynomial functions? Well yes, as Karl said, but it depends somewhat on your point of view. I may not think the regular function 1/z is a polynomial function on the set {z≠0} in A^1. But to understand Karl’s point, I must realize that z is not an affine coordinate system for that subset, so I should not expect all regular functions there to be polynomials in z. But the pair of functions (z,1/z) = (z,w) is an affine coordinate system there, and then 1/y is a polynomial in w, i.e. 1/y is a polynomial in the variable 1/y! In this way one can make any regular locally rational function look locally polynomial. Here are some details as I understand them.

Define an abstract variety as a topological space with a basis of open sets {Uj} such that each Uj is equipped with a homeomorphism fj:Uj-->Vj where Vj is a Zariski closed subset of some affine space. Then require that every inclusion map Ui into Uj becomes a polynomial map of the corresponding affine varieties (fj)o(fi)^(-1):Vi-->Vj.

In this category Donu’s definition of morphism makes perfect sense. I.e. a continuous map of abstract varieties is a morphism if it is locally polynomial in some collection of coordinate systems. In particular a continuous k valued function is regular if and only if it is locally defined by polynomials in some coordinate cover. Nothing could be conceptually simpler or more natural.

What is the catch? With this definition it is not immediately obvious that any familiar (non finite) example at all is an abstract variety, not even k ≈ A^1. I.e. it takes an infinite number of affine coordinate charts even to give affine space itself the structure of an abstract variety. Moreover these charts are defined by rational functions.

If we define a coordinate system in U as a finite set of regular functions such that every regular function in U is a polynomial in terms of those functions, then it is sufficient but not necessary to be affine in order to possess a coordinate system. Fortunately every affine variety has a topological basis of affine open sets, thus we can put a structure of abstract variety on every quasi projective variety.

The difference between the algebraic case and the analytic and differential cases is that the restriction of an affine coordinate system may not be an affine coordinate system. Thus we cannot use the same coordinate system on every open subset of affine space, as we would in differential geometry.

Lemma: The principal open subsets Uf = {f≠0} define a structure of abstract affine variety on affine space A^n.
Proof: It suffices to use polynomials f which have no repeated prime factors. Define the coordinate map Uf-->A^(n+1) by sending x-->(x,1/f(x)) = (x,T). Then the image Vf = {1-f.T = 0} is a closed affine set. The coordinate map itself is defined by regular rational functons on Uf, and is a homeomorphism.
Moreover, Ug is contained in Uf if and only if g = fh, for some polynomial h. If we map Ug-->Vg by x-->(x,1/g(x)) = (x,W), then Vg = {1-g.W=0}. Hence the inclusion map from Ug to Uf becomes in affine coordinates, the map (x,W)-->(x,W.h(x)) from Vg to Vf, a polynomial map in the coordinates (x,W). QED.

With this lemma it seems one can use restrictions to define a structure of abstract variety on every quasi affine and quasi projective variety.

Finally, as Donu remarked, it is not obvious that one can check regularity using any coordinate cover. I.e. there might be one cover by affine coordinate systems in which a given map is locally polynomial, and yet another cover by different affine coordinate systems in which it is not. One must prove the usual theorem, via the nullstellensatz, that a locally polynomial function on an affine variety is globally polynomial.

This is a beautiful, conceptually natural point of view on what it means to be a morphism of varieties. I would advocate, after giving this definition, proving that a map of quasi projective varieties is a morphism in this sense if and only if it has the property in the accepted answer above, if and only if it can be defined locally by sequences of homogeneous polynomials with no common zeroes. I.e. it is hard to be prepared for all situations with just one characterization.

The last (homogeneous polynomial) point of view can then lead to the important idea that a morphism of a variety to projective space is also defined by a line bundle and a sequence of regular sections without common zeroes, an approach not yet mentioned. I.e. maps of a variety to projective space are much more special than maps to arbitrary varieties, and this special case is well worth understanding.

If we want to discuss the meaning of the common zeroes of sections, as in the example of the intersection curve of two quadrics, note the restriction of a linear polynomial to this curve must vanish 4 times, so the image of the map defined by linear polynomials should have degree 4. Since the image curve has degree three there must be a point where the rational map is not defined. I.e. the line bundle on the intersection curve defining the morphism is O(1) restricted to the curve tensored with O(-p) where p is the point [0:0:0:1]. Since we extended the map using quadratic polynomials, each vanishing 8 times on the curve, presumably those polynomials have 5 common zeroes on our curve.