Reading Sheaves on Artin stacks by M. Olsson I find this definition (3.7, (i)):
Let $\mathcal X$ be an algebraic stack on a scheme $S$. A sheaf of rings $\mathcal A$ on $\mathcal X_{\textrm{lis-et}}$ is flat if for any smooth morphism $f \colon U \to V$ in $\operatorname{Lis-Et}(X)$, the natural map of sheaves on $U_{\textrm{et}}$ $$ f^{-1}(\mathcal A_V) \to \mathcal A_U $$ is faithfully flat.
On the other hand, I read the book Champs algébriques by G. Laumon and L. Moret-Bailly, and in (12.7) there I find the following definition (I translate and adapt a little):
A sheaf of rings $\mathcal A$ on an $S$-algebric stack $\mathcal X$ is flat if for any arrow $\varphi \colon U \to V$ in $\operatorname{Lis-Et}(X)$ with $\varphi$ smooth, the homomorphism of étale rings $$ \varphi^{-1}\mathcal A_V \to \mathcal A_U $$ is flat.
Now, I must be missing something, but why does the first definition require faithful flatness whereas the second one just flatness? Are those definitions actually equivalent for some reason I can't figure out?
The book by Laumon and Moret-Bailly goes on to say (Examples 12.7.1) that the structure sheaf $\mathcal O_{\mathcal X}$ on an algebraic stack $\mathcal X$ is indeed flat. Does this hold if we assume the first definition of "flatness"? (I certainly hope so...)
