Is this variation of set-cover NP-hard to approximate?
Your first problem is a classical NP-hard problem known as maximum coverage. The greedy algorithm gives a $1-1/e$ approximation, and this is tight (assuming P≠NP).

Your second problem is a special case of set cover. Indeed, take any instance of set cover, and add to it the set $S$ itself. If the optimal solution for the set cover instance is $M$, then the optimal solution for the new instance (of your problem) is either $M$ or $M+1$. (You can also add $S \cup \{x\},\{x\}$, where $x$ is a new element, to make the optimal solution of the new instance exactly $M+2$.) Hence your problem inherits the $\ln n$-hardness of set cover. In the other direction, the greedy algorithm likely gives an $O(\log n)$ approximation for your problem.