The notion of hypercovering (or hypercover) was introduced by J.L. Verdier in SGA 4. In his definition, a hypercovering $K_\bullet$ is a semirepresentable semisimplicial presheaf on a site $C$ such that the canonical morphism $K_{n+1}\to cosk_n(K_{n+1})$ is a "covering isomorphism of presheaves" for $n\geq 0$ and $K_0\to 1$ is a covering morphism ($1$ is the final object in $Psh(C)$).
This should be more or less the same definition used by Moerdijk in Classifying spaces and classifying topoi, page 17, while treating Grothendieck topoi: a hypercover(ing) in a topos $\mathcal E$ is a simplicial object $X_\bullet$ in $\mathcal E$ such that the map $X_\bullet\to 1$ is a "local trivial fibration" in the sense of Quillen, Homotopical algebra.
Now I have trouble in connecting these two definitions with the idea of a hypercovering in the \'etale site of a scheme $X$: this should be a "generalised covering", so a simplicial étale scheme $U_\bullet$ s.t. $U_0\to X$ is a covering and for every $n$ the canonical morphism $U_{n+1}\to cosk_n(U_\bullet)_{n+1}$ is a covering. Here a covering is just an étale surjective scheme on $X$.
So the point is: the definition by Verdier and Moerdijk is at the level of (pre)sheaves, i.e. of topoi, while the special definition in the étale context seems to be at the level of sites. It is actually true that in Verdier definition everything is (semi)representable, so it wouldn't seem so strange that, in fact, we are speaking at the level of the site. But this doesn't seem so clear in the definition used by Moerdijk, which is actually the most useful in order to treat topoi.
Thank you in advance for any help against this confusion of mine.
The Yoneda embedding sends schemes to representable presheaves and prserves limits, so a hypercover in the third sense gives rise to a hypercover in the first sense.
The sheafification functor sends presheaves to sheaves (i.e., objects in the underlying topos) and preserves finite limits (which occur in the coskeleton construction), so a hypercover in the first sense gives rise to a hypercover in the second sense.
Finally, any Grothendieck topos T is equivalent to the topos of sheaves on T equipped with its canonical topology, so a hypercover in the second sense gives rise to a hypercover in the first sense.