Hamiltonian path – Prove that circulating regular regular graphs of degree at least three contain all even cycles.

That's the question I'm trying to solve, but in my research on the flowing graph, I discovered Paley's 13th order graph.

Now, clearly, when we look at this graph, which is an example of a graph running at a cycle of 13 lengths, which is odd, we can therefore refute the instruction given above.

There was also a proof that I came across, that I could not collect much, This is the link of the evidence. In this, they prove that it is true, for an odd number of vertices in Lemma 2.
I am very confused now, I miss something, is there any other interpretation of this question? My interpretation is that if there is a cycle in the flowing graph, it must be of equal length.

Any idea would be useful.