How do we define a tree in a directed graph?
Sometime there isn't a completely agreed upon meaning of terms, it is more useful to look at the context to see which definition is appropriate.

In this case, instead of searching for definition of general directed tree, it is better to look for directed steiner tree.

A quick search pulled up this paper which has this diagram showing an example of directed steiner tree:



While not explicitly defined, it coincides with my intuition of what a default directed tree should look like --- A directed graph with exactly one directed path from a distinguished root vertex to any other vertex in the graph (tree).