OK. I found several definitions of a geodesic cycle in the literature. Hopefully, they are all equivalent, and they are equivalent to the one used in the FSP page, which tries to look less "hard-math", but maybe at the price of being unclear.
"A finite cycle C in a graph G is called geodesic if, for any two vertices x, y ∈ C, the length of at least one of the two x–y arcs on C equals the distance between x and y in G."
"A cycle in a graph is geodesic if the distance of each pair of nodes on the cycle coincides with their distance restricted on the cycle"
Seems to me none of those is crystal-clear. If I was to put it in "hard-math" language, I would put it that way.
Let G be a graph and C a cycle in G. Let's call dG(x,y) the distance of two nodes in G, and dC(x,y) their distance in the induced subgraph G[C]. The induced subgraph is a graph containing all the nodes of C, and all the edges joining nodes of C in G.
C is geodesic if and only if for any pair (x,y) of nodes of C, dC(x,y)=dG(x,y)
See
https://en.wikipedia.org/wiki/Induced_subgraph
If the cycle is (v1,v2, ...,vn) of length n, we can compute dC from the values of indices in the cycle.
dC(vj,vk) = Min( |k-j|, n-|k-j| )
We can go on frightening people away. The intuitive definition "a geodesic cycle is a cycle with no shortcut inside" seems more suitable for the casual WikiTreereader.
To your question : if there is an alternate path of same length between two antipodes, is the cycle still geodesic? I would say yes. Suppose you have a geodesic cycle of legnth 6 (v1,v2,v3,v4,v5,v6), and an alternate geodesic path (v1,w2,w3,v4) from v1 to its antipode v4. Then we have three distinct geodesic cycles with antipodes v1,v4, the original one plus (v1,v2,v3,v4,w3,w2) and (v1,w2,w3,v4,v5,v6)
For the rest : I've put the 2-cycles out of the definition, because the edges are not distinct. In other words, I want the cycles to be Eulerian (each edge is traversed only once).
The other propositions seem correct. Not sure checking what you propose would shorten of complexify the algorithm.