Since I seem to have some "intuition" about what "should" be a digraph-minor and what "shouldn't", I decided to test my "intuition" on some digraphs. A minor is actually the composition of taking a subgraph and "taking an M-graph". Because there is no confusion what is meant by "taking a subgraph", only "taking an M-graph" needs clarification.
In the picture below, the digraph on the right is an M-graph of the digraph on the left (and of the digraph in the middle), but the digraph in the middle is not an M-graph of the digraph on the left (according to my "intuition", because $a2$ should not be reachable from $a3$).
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Edit 1 It may be a good idea to restrict contractions to weakly connected subsets, so I slightly modified the definition of an M-graph (which is intended to formalize my "intuition"):
A digraph $G'=(V',E')$ is an M-graph of a digraph $G=(V,E)$, if there is a surjective function $\varphi:V \mapsto V'$ such that
- $\varphi^{-1}(\{v'\})$ is a weakly connected subset of $G$ for all $v'\in V'$,
- every walk on $G'$ is the image under $\varphi$ of a walk on $G$, and
- every image under $\varphi$ of a walk on $G$ is a walk on $G'$.
Here a walk on a digraph $G=(V,E)$ is a sequence $v_0, v_1, \ldots, v_n$ of vertices from $V$ such that $(v_i,v_{i+1})\in E$. The image of a walk under a function $\varphi$ is the sequence of the images of the vertices of the walk, where identical successive vertices have been replaced by a single vertex.
This gives the following definition of a minor for a digraph:
A digraph $X$ is a minor of a digraph $Y$, if $Y$ contains a subgraph $Z$ for which $X$ is an M-graph.
There are cases where this definition allows more minors than my "intuition":

There are also cases where all subsets must be contracted at the same time, showing that incremental constructions contracting one edge or subset at a time won't work (in general) for this definition of digraph minor:

It should be possible to prove some useful properties for this definition of digraph minor. However, this doesn't necessarily answer the question whether this definition of a minor of a digraph is actually useful.
Edit 2 The above minor relation is not a well quasi ordering, as shown by the following counterexample:

Maybe it's not such a good idea after all to restrict contractions to weakly connected subsets. But even without this restriction, it's unclear whether the corresponding minor relation would be a well quasi ordering.