In SGA1 Expose 13 it says: (I'm translating) Let $S$ be a scheme, and $f: X \rightarrow Y$ a morphism of $S$-schemes. Definition 1.1: Let $F$ be a stack over $X$. We will say that $(F,f)$ is cohomologically proper relative to $S$ in dimension $\leq -1$ (resp. in dimension $\leq 0$, resp. in dimension $\leq 1$) if, for every $S$-scheme $S'$ ($g:S'\rightarrow S$), the canonical functor: $g^*f_*F \rightarrow f'_*h^*F$ (where $h$ is the canonical $h: X\times_S S' \rightarrow X$ and $f'$ is the base change of $f$ to $S'$) is faithful (resp. faithfully flat, resp. an equivalence of categories).

This seems to suggest that they had a more general idea in mind of what "cohomologically proper" means. What is it?

In SGA1 Expose 13 it says: (I'm translating) Let $S$ be a scheme, and $f: X \rightarrow Y$ a morphism of $S$-schemes. Definition 1.1: Let $F$ be a stack over $X$. We will say that $(F,f)$ is cohomologically proper relative to $S$ in dimension $\leq -1$ (resp. in dimension $\leq 0$, resp. in dimension $\leq 1$) if, for every $S$-scheme $S'$ ($g:S'\rightarrow S$), the canonical functor: $g^*f_*F \rightarrow f'_*h^*F$ (where $h$ is the canonical $h: X\times_S S' \rightarrow X$ and $f'$ is the base change of $f$ to $S'$) is faithful (resp. fully faithful, resp. an equivalence of categories).

This seems to suggest that they had a more general idea in mind of what "cohomologically proper" means. What is it?