When I am studying stacky stuffs, I am always confused by the notion of étale abelian sheaves on $X$, because conceivably there might be three different meanings of that:
Take the global étale site (the category of scheme with topology defined by étale coverings) and consider an abelian sheaf $\mathcal F$ on the global site, together with a map $\mathcal F\rightarrow h_X$ to the sheaf represented by $X$.
Take the big étale site on $X$ (the category of schemes over X with topology defined by étale covering) and consider an abelian sheaf on this site.
Take the small étale site on $X$ (the category of schemes étale over X with topology defined by étale covering) and consider an abelian sheaf on this site.
From the context of the books that I am reading, (1) doesn't seem like the right thing to look at. But is (2) or (3) the right thing to look at? Are they the same?
I don't think 1) is correct. In the notes that I have on stacks (some notes by Martin Olsson) we normally stick to 2) sheaves on the big etale site, I guess because its slightly more general and because:
a) you can prove that $X$-schemes give you sheaves on the big Etale site (or more generally on the fppf site).
b) Algebraic spaces are defined as sheaves on the big etale site satisfying some additional properties.
But I think that your definition 3) would be equally acceptable in other contexts.