Does 'finite + finitely presented as an algebra' equal 'finitely presented as a module'?

EGA IV$_1$, 1.4.7.

It appears that in the meantime, full proofs have been added to the Stacks Project.

Tag 0564. Let $R \to S$ be a finite and finitely presented ring map. Let $M$ be an $S$-module. Then $M$ is finitely presented as an $R$-module if and only if $M$ is finitely presented as an $S$-module.

In particular, $S$ is finitely presented as an $R$-module if $S$ is finitely generated as an $R$-module and finitely presented as an $R$-algebra.

Tag 058R. Let $M$ be an $R$-module. Then $M$ is finite projective if and only if $M$ is finitely presented and flat.