Definition of k-set cover I'm trying to understand the sparsification lemma by Impagliazzo, Paturi and Zane (IPZ) (from this article) and in their proof they reduce the k-SAT problem to the k-set cover problem. But their definition of the problem seems to be different from the definition found in other articles where the problem is simply the normal set cover with the addition that each set is limited to contain a maximum of k elements from the universe, as can be seen in eg. this description from Führer and Yu (taken from this article):



The solution to this problem then consists of a collection of sets whose union covers the universe. But IPZ states the problem in this way:



and in that, the solution is defined as a set C that contains at least one element from each of the sets S in $\mathscr{S}$. It is not stated that the collection of sets necessarily covers the universe. Finding a C of size |$\mathscr{S}$| is therefore trivial and finding a minimal C corresponds to finding elements that overlap different sets from $\mathscr{S}$ in the best possible way.

After this lengthy intro my question is this: Am I somehow not reading the definition given by IPZ correctly and is it just an other way of stating the same problem? Or is this two different problems that unfortunately has been given the same title?
The second definition uses the hitting set formulation, which is equivalent to the set cover problem. To see that, you may reverse the roles of sets and elements.  You can find more information on the wikipedia page.