Computing the edge orbits of a graph (and discussing definitions)
It turns out that yes, indeed, there are two possible definitions for edge-automorphisms... but it turns out that they almost always coincide so that it seems that people often get away with not making the distinction. 

First, some notation. For a simple graph $G = (V,E)$ we let $\Gamma_V(G)$ define the group of automorpisms over the set of vertices $V$ that preserve $G$'s edge structure. Similarly, we define $\Gamma_E(G)$ to be the group of automorphisms over the set of edges $E$ that preserves $G$'s edge incidence structure. 

For any vertex-automorphism $\pi \in \Gamma_V(G)$, we observe that there is a natural edge-automorphism $\pi':E \to E$ induced by $\pi$. Let $\Gamma'_V(G)$ denote the group of such induced edge-automorphisms.


  Theorem: $\Gamma'_V(G)$ is isomorphic to $\Gamma_E(G)$ for all simple, connected $G$ with three or more vertices, with three exceptions: $K_4$, $K_4$ with one edge deleted, and the graph illustrated above within the question.  


In other words, the example presented in the question happens to be on one of the very few graphs that make the distinction between $\Gamma_E(G)$ and $\Gamma_V'(G)$ important. Also, it is trivial to observe that this isomorphism does not hold for graphs with two or fewer vertices.

I believe that the canonical reference for this result is [1], though the proof is somewhat complicated with several cases. There are references to some more streamlined proofs of this result in [2], but I haven't tracked those down.


Whitney, Hassler. "Congruent Graphs and the Connectivity of Graphs." American Journal of Mathematics 54.1 (1932): 150-168.
Hemminger, R. L. "On Whitney's line graph theorem." The American Mathematical Monthly 79.4 (1972): 374-378.