How to call covers not covering anything else?

Let $A$ be a set. Then it seems common to name cover of $A$ any set $\mathcal C$ of sets with $\bigcup{\mathcal C}\supseteq A$.

1. Is there then a good name for the particular covers $\mathcal C$ with $\bigcup{\mathcal C}=A$ ?

2. Are there standard names for those covers $\mathcal C$ with $\emptyset\notin\mathcal C$ ?

Question 2 also arises for partitions. Well-regarded books allowing the empty set to be a member of a partition (which I then would only call a quasipartition) but using the term differently later on include Kelley's and Dugundji's topology texts.

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The construction "covering of $A$ by [something]" seems handy. For (1), one can use "covering of $A$ by subsets of $A$", and for (2) one can use "covering of $A$ by nonempty sets".
An answer to the first question is: An exhaustion of $A$.