Comment by Gro-Tsen on Equivalence of definitions of locally finitely generated sheaves

That $\mathcal{F}$ is “generated by the $s_i$” means precisely that the morphism $\mathcal{A}^{\oplus n} \to \mathcal{F}$ is surjective (see, e.g., EGA 0.5.1.1). This is often expressed, instead, as a condition on the stalks: that every $\mathcal{F}_x$ is generated by the germs of the $s_i$ at $x$. This makes your equivalence trivially true, but it remains a legitimate question whether the equivalence with your understanding holds (I'm pretty sure not, but I don't have a counterexample at hand).