When talking about the length of a path in a graph, what exactly is a skip?
The second definition that you quote is informal, and open to interpretation. Fortunately, the same paper On $k$-skip shortest paths by Tao, Sheng and Pei contains a formal definition, Definition 2 on page 4.

That said, $k$-skip is not a common term, and doesn't really have a universally accepted definition (or indeed, any universally known definition), except perhaps in work following up on the Tao et al. paper mentioned above.