My search has not turned up anything. Take a directed graph which satisfies that for any two vertices of the graph, there is at most one path connecting them. Is there a term for these sorts of graphs?
Such directed graphs have been called singly connected (for an example of this usage, see this paper).
But you should be careful, and define this term before using it, because other sources (such as Wikipedia) use the term synonomously with a polytree: a directed graph whose underlying undirected graph is a tree.