What is an algebraic map?

Question

From Milne's Algebraic Varieties over Complex Numbers note:

COROLLARY 15.6. Any holomorphic map from one projective algebraic variety to a second projective algebraic variety is algebraic.

PROOF. Let $\varphi:V\rightarrow W$ be the map. Then the graph $\Gamma_{\varphi}$ of $V$ is a closed subset of $V\times W$, and hence is algebraic according to the theorem [Chow's Theorem]. Since $\varphi$ is the composite of the isomorphism $V\rightarrow \Gamma_{\varphi}$ with the projection $\Gamma_{\varphi}\rightarrow W$ and both are algebraic, $\varphi$ itself is algebraic.

I found the following definition from planetmath

A map $f:X\to Y$ between quasi-affine varieties $X\subset k^n,Y\subset k^m$ over a field $k$ is called algebraic if there is a map $f':k^n\to k^m$ whose component functions are polynomials, such that $f'$ restricts to $f$ on $X.$

Alternatively, $f$ is algebraic if the pullback map $f^*:C(Y)\to C(X)$ maps the coordinate ring of $Y$, $k[Y],$ to the coordinate ring of $X$, $k[X]$.

But it didn't make much sense in the above context. What is the definition of an algebraic map in the context of this corollary?

Answer 1

Milne points to Shafarevich 1994, Theorem 15.5. On p. 177, Shafarevich proves a more general result

Theorem 8.5. If $X$ and $Y$ are complete algebraic varieties then any holomorphic map $f : X_{an}\rightarrow Y_{an}$ is of the form $f = g_{an}$ where $g : X \rightarrow Y$ is a morphism.

So, algebraic in the stated corollary refers to a morphism of algebraic varieties as Dietrich Burde pointed out in the comments.