Simple cycles of length two in an undirected graph
It turns out that the very popular textbook Introduction to Algorithms by Cormen, Leiserson, Rivest, and Stein, sneakily avoids this issue by giving a different definition of cycle in undirected vs directed graphs.

In Section B.4, it defines cycle and simple cycle as I do (they also require at least one edge), in directed graphs. In an undirected graph, they define cycle the same way except they require that all of the edges in the cycle be distinct; and they say the cycle is simple if in addition the vertices are distinct, other than $v_1=v_k$.