Could you explain in what ways was the Russian translation of Hurwitz-Courant an improvement over the original?

I think both of you are right. The formalism is somewhat implicit in his works on coherent duality, but wasn't explicitly presented until one of the first oral exposés of SGA 5. This exposé, according to Récoltes et Semailles, was split, its contents distributed between those (published) exposés of SGA 4 and the first two (published) exposés of SGA 5, the second one of which was instead put in Deligne's "Cohomologie étale".

I recall that the name "tannakian" was, un fact, due to Grothendieck himself, but he didn't realise it until the end of Récoltes et Semailles. It's true that he preferred the other name you mention.

I think that topos were defined precisely so that cohomology had a sense for them... This seems a rather vacuous assertion.