Vertex cover of a graph by removing leaf-vertices from a DFS tree
Look at the definition of vertex cover (as provided by the book). It is strictly defined on undirected graphs. Thus, the answer doesn't apply to directed graphs, nor to any other kinds of graphs you might think of. So your counterexample is invalid, as it makes invalid assumptions (graphs are always undirected here). These invalid assumptions are the source of your confusion.

To consider directed graphs, we first need to define what a vertex cover is on a directed graph. Well, we can say it is a subset of vertices such that every arc is incident to at least one vertex in the subset. Again, the book makes no claim about such directed vertex covers. If you are now asking "does $\{ a \}$ form a (directed) vertex cover as per our definition?", the answer is NO as you correctly observed.