Module-finite affine maps for the quasi-finite comparison

For affine classical algebraic sets $X,Y$, call a morphism $f:X\to Y$ module-finite if $k[X]$, via pullback, is a finitely generated $k[Y]$-module. This is the affine module criterion. Empty affine sets are allowed, with zero coordinate ring; the definition does not assert a global affine-preimage criterion for arbitrary varieties.

Work over a fixed algebraically closed field $k$, with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.