@antoinechambertloir@boarders@joshuagrochow@MartinEscardo - there is an amusing interplay between the words "cover" and "covering" in mathematics. As I started learning topology I thought an "open cover" was something completely different from a "covering space" - indeed, strangely, it never occurred to me to wonder if these concepts were related. But then, learning concepts like "etale map" and "etale topology", and learning that an open cover (U_\alpha) of a space (X) gives a local homeomorphism (\sum_\alpha U_\alpha \to X), I realized that open covers and covering spaces are deeply related. Of course this is something that all experts already know.... but I became curious why people don't run around saying "hey! coverings and covers both give etale maps!"
I have no idea what Serre and Grothendieck and others thought about "covers versus coverings" - some of this may be special to English-language math - but Serre's observation that "algebraic fiber spaces" can be trivialized by pulling back along an étale covering sounds like exactly the right reason to develop algebraic geometry in a way that emphasizes their relationship.