Consider a topological space $X$, and an open cover of $X$. Now when choosing a Grothendieck topology we select "coverings", which in the case of $\mathcal O(X)$ should be open covers. But not every "cover" is a "covering", i.e. the disjoint union if the covering open sets need not form an unramified covering.
So why use the word "covering" and not just "cover" for Grothendieck topologies? Are there historical reasons? Or is there something wrong in my argument above?
Thank you in advance.
The historical reason is that these terms were translated from French: a cover is un recouvrement and a covering is un revêtement.
This is one of the rare cases where French is less ambiguous than English.
For example the map $ \:\mathbb S^1\to \mathbb S^1:(\cos t,\sin t)\mapsto (\cos 2t,\sin 2t)$ is un revêtement de $\mathbb S^1$ and definitely not un recouvrement de $\mathbb S^1$.
Conversely the set of all open intervals $(n,n+2)\subset \mathbb R \;(n\in \mathbb Z)$ is un recouvrement de $\mathbb R$ and definitely not un revêtement de $\mathbb R$
Also note that the map $\mathbb C^*\to \mathbb C^*:z\mapsto z^2$ is both un revêtement étale de $\mathbb C^*$ and un recouvrement pour la la topologie étale de $\mathbb C^*$ : one has to keep a clear mind here!