ag.algebraic geometry - Affine morphisms in different settings coincide?

Emerton and Zoran's answers completely answer the question as stated, but there's another way to think about affine morphisms that is worth mentioning.

Given any quasi-compact and quasi-separated morphism of schemes $f:Xto Y$, $newcommand{O}{mathcal O}f_*O_X$ is a quasi-coherent sheaf of $O_Y$-algebras. This functor has an adjoint, called relative $Spec_Y$, or relative Spec. Given a quasi-coherent sheaf of $O_Y$-algebras $newcommand{A}{mathcal A}A$, we get a scheme over $Y$, $phi^A:Spec_Y Ato Y$, with the property that $phi^A_*(O_{Spec_Y A})=A$ and $Hom_Y(X,Spec_Y A)cong Hom_{O_Ytext{-alg}}(A,f_*O_X)$ for any $f:Xto Y$. A morphism $f:Xto Y$ is affine if and only if $Xcong Spec_Y(A)$ (as a $Y$-scheme) for some $A$ (which must be $f_*O_X$). See EGA II §1 for this development of affine morphisms.

I find this way of thinking about affine morphisms is useful for two reasons. First, if I'm working with a bunch of schemes affine over $Y$, it's often easier for me to think about a bunch of $O_Y$-algebras with algebra morphisms between them. Second, the adjunction in the previous paragraph tells you that any (quasi-compact quasi-separated) morphism $f:Xto Y$ has a canonical factorization through an affine morphism $Xto Spec_Y(f_*O_X)to Y$, called the Stein factorization (the first morphism is Stein, meaning that the structure sheaf pushes forward to the structure sheaf). This factorization is often extremely handy; for example, if a morphism is quasi-affine, the Stein factorization is a witness of its quasi-affineness.