Is Directed Graph a Graph?
As is often the case, using a formal definition is helpful:

Let $V$ a finite set. $G=(V,E)$ is


a graph if $E \subseteq \left\{\{v_1, v_2\} \mid v_1, v_2 \in V \right\}$ and 
a digraph if $E \subseteq \left\{(v_1,v_2) \mid v_1, v_2 \in V\right\}$.


Note the central difference: edges are sets in graphs and pairs in digraphs. In particular, simpleness is implied by this definition. Extending the definition is also easy:if $E$ was a multiset, you could have non-simple graphs. If the edges had more than two components, you'd have hypergraphs.

Disclaimer: People define (di)graphs in different ways; this is one very common variant. For example, if you are uncomfortable with digraphs (formally) not being graphs, you define them like this:

Let $V$ a finite set and $E \subseteq V^2$. We call the pair $G=(V,E)$ a graph. We say 


$G$ is undirected if and only if $(v_1,v_2) \in E \Longleftrightarrow (v_2,v_1) \in E$ and
$G$ is directed otherwise.


This defines undirected graphs as special cases of directed graphs. Note that with this definition, extensions to labeled graphs (edges get markings) may be awkward: We want the complete digraph with to be different from the complete undirected graph (as the former has two labeled edges between between every pair of nodes, the latter only one); by this definition, they are the same. Note how the first definition I gave circumvents this issue nicely; sometimes definitions are (re)made with later needs in mind.