Don't see why we need to forbid self-links. The truly interesting question is "What is the longest cyclical graph". It may be that the answer is 1, but hopefully there's something longer out there.

You can see it that way if you like, but I don't want to see a self-link being used to pad out the length of a nontrivial cycle!

Can we just say that a band can’t appear more than once in a cycle? Depends whether you see figure-8 cycles as cheating, I suppose.

I think it's a question of improbability: the impressiveness of any result is directly related to its unlikelihood of appearing on the assumption that graph edges appear at random with an appropriate probability distribution.

Self-links are known to be common (eponymous albums and/or songs). Hence the base probability model on which to judge impressiveness should reflect that, by considering self-links to appear with a significantly larger probability than non-self links. So including a self-link in a cycle doesn't increase its impressiveness by very much (so I'd be inclined to consider it a cheap trick), and finding a cycle entirely composed of one self-link is definitely too easy to be a challenge. But two nontrivial cycles sharing a vertex would be a more impressive find, so I'd accept the resulting figure-8 as a worthwhile result.