In "stack project", there is a lemma on finite locally free morphism, 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 finite 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 statement is equivalent?

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

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?

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