If you want a good definition of edge you can review books on "Graph
Theory". In fact, the use of edge in networks comes from there. Hence, the
definition of "edge" is not more than the connecting line between "nodes"
(vertices of the graph). End-to-End connections can be hold
nodes using multiple "edges".
A graph edge and a network edge are two different things! But if you prefer
"edges" to "links" than you should use "vertices" instead of "nodes" (see
your last sentence). The term "network edge" comes from the dictionary and
has nothing to do with graph theory.