What is the difference between directed graph and bidirectional graph?
As stated in the Wikipedia article, a bidirected graph is a graph where all edges have an orientation at both edges of the edge. The orientation is an arrow head. So in your example, it is not a bidirected graph because the edges only have orientation on one end.
The following would be an example of a bidirected graph. Notice there is an arrow head on both sides of each edge of the graph.

There are three types of edges of a bidirected graph.

Extroverted edge: arrow heads oriented outwards or towards vertices (e.g. $\{a,b\}$)
Introvert edge: arrow heads oriented inwards or away from vertices (e.g. $\{b,c\}$)
Directed edge: arrow heads oriented in one direction or away from one vertex and towards the other (e.g. $\{a,c\}$)

There are also loose edges (which have no orientation on either end) and half edges (which have orientation on one end).
Another use of bidirected graph is a directed graph where each edge has a matching edge going the opposite direction. Your example would not be a bidirected graph in this context either since there are edges that do not have the reverse direction (e.g. $(1,3)$ exists but $(3,1)$ does not). The following is an example of bidirected graph in this context.

Bidirected graphs are not common in my experience. This is just from a quick read of the Wiki article.

Bidirected graphs: https://en.wikipedia.org/wiki/Bidirected_graph
Directed graphs: https://en.wikipedia.org/wiki/Directed_graph