good point - but if you actually search for "rabbit chasing dog", you do get a rabbit chasing a dog video - https://www.youtube.com/watch?v=GBYdaPacHJg, as well as some for the other way around

In this case, could you even know which one was actually meant?

Technically, I'd think that the ambiguity is removed when you use "rabbit-chasing dog", but that is assuming the text available is sufficiently precise.

I’ve always seen that hyphenated: “rabbit-chasing dog”.