Wikipedia gives the path cover definition as:
Given a directed graph $G = (V, E)$, a path cover is a set of directed paths such that every vertex $v \in V$ belongs to at least one path.
I'm reading the paper "On k-Path Covers and their Application" (Funke, S., Nusser, A., & Storandt, S. (2014)) and they gave the definition where we select a subset of vertices $C \subseteq V$ such that for every simple path $\pi$ in the graph we have that $C \cap \pi \neq \emptyset$. (There's also a $k$ constraint but it's not particularly relevant to my problem).
Are these definitions equivalent? One uses a set of vertices and the other uses a set of paths.