TITLE: Does 'finite + finitely presented as an algebra' equal 'finitely presented as a module'?
QUESTION [6 upvotes]: In "stack project", there is a lemma on finite locally free morphisms, saying that a finite locally free morphism of schemes is equivalent to a morphism which is finite, flat, and locally of finite presentation.
For the proof, they refer to the commutative algebra fact that a module is finite locally free iff it is flat and finitely presented. 
In order to complete the proof, I have the following gap:
Let $A \to B$ be a morphism of rings. Then $B$ can either be viewed as an $A$-algbra or an $A$-module. Is it true that the following two statements are equivalent?

$B$ is a flat $A$-module and is finitely presented as an $A$-module.
$B$ is a finitely generated flat $A$-module, and is finitely presented as an $A$-algebra.

REPLY [3 votes]: 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.