If B is a cover of A, then do we say that C is a subcover of B, or of A?

If B is a cover of A, then do we say that C is a subcover of B, or of A?

My textbook variously says both "subcover of $A$" and "subcover of $B$" to
refer to a subcollection $C$ of the collection $B$ (that covers $A$); is
this usage standard and is it not potentially confusing?
Example sentence: "Consider the open cover $B$ of $A$. By Lemma 2, $B$ has
a finite subcover $C$ of $A$." I appreciate that it is useful to be able
to specify the "base" set $A$, but isn't $C$ a subcover of $B$, instead of
a subcover of $A$?
Another question: are these three sentences equivalent (A and B are sets)?
$B$ covers $A$.
$B$ contains $A$.
$A$ is a subset of $B$.