Does a "song" have to involve "singing"?

If it works for you, go for it. The thing that’s getting obvious is that “song” is a highly personalized hint at something to do with music. To use it with more precision than that means you had better be sure how the other person/people you’re talking with see the term.

Might as well call it music-thingy.

I just take it a little further. Instead of just words, there must be vocalization, either human or animal.