The first paragraph of the section Overview in the paper Principal infinity-bundles - General theory by Nikolaus, Schreiber and Stevenson https://arxiv.org/abs/1207.0248 precisely reads the following:

The concept of a G-principal bundle for a topological or Lie group G is fundamental in classical topology and differential geometry,. More generally, for G a geometric group in the sense of a sheaf of groups over some site, the notion of G- principal bundle or G-torsor is fundamental in topos theory. Its relevance rests in the fact that G-principal bundles constitute natural geometric representatives of cocycles in degree 1 nonabelian cohomology H1(−,G) and that general fiber bundles are associated to principal bundles.

My confusion is the following:

Since they are working with a sheaf of groups $G$ over a site(which they call here geometric group) so it is natural to ask what did they mean by $G$ Principal bundle or $G$-torsor in this context(Note neither they have mentioned about the "base space" of the $G$ Principal bundle nor about where $G$-torsors are defined(like over a topological space or a site?).

Though I am assuming $G$ torsors are defined over a site. But do they mean that the notion of $G$ principal bundles and $G$ torsors over a site are indistinguishable in their set up?

Or

Is there any separate meaning for $G$ Principal bundles for them?

Also they said $G$ Principal bundles constitute natural geometric representatives of cocycles in degree 1 nonabelian cohomology H1(−,G) . Note instead of writing $H^1(X,G)$ they wrote $H^1(-,G)$. So my guess is if $B$ is the base space of the $G$ Principal bundle then an isomorphism class of such $G$ principal bundle over $B$ will correspond to an element of $H^1(B,G)$.

Thank you.