A Grothendieck topos can be defined either as the category of sheaves on a site, or as a category satisfying Giraud's axioms, or as an elementary topos that is bounded over $\rm Set$. I believe any of these definitions can be taken as basic and a good deal of theory developed before proving the equivalence to the others.
For that matter, a sheaf on a topological space (or locale) $X$ can be defined either as a presheaf satisfying a gluing condition or as a local homeomorphism with codomain $X$.