Foreign keys are put in entity tables in normalized databases.
> The identity of neither the pet or the owner depend on each other
So then the only thing that belongs on the pet itself is an arbitrary id? Because the identity of the pet doesn't depend on it's age or when it was born, and a timestamp is just a timestamp, so putting "time of birth" could just be a link relationship to another entity.
Same for breed. Location too, that's not a tool for identifying the pet. Heck, even the name is often defined by the pet-owner relationship but not the pet itself.
So we're left with a pet object that has only a uuid, and a set of link relations in the headers.
Why is that better? You've said my way is "wrong", but you haven't actually explained what your way does to improve the situation. I just demonstrated that any arbitrary piece of info can be represented as a link relationship, so the whole thing is arbitrary anyway.
So the simple question, the only question is, as a designer or user of an API, how does sticking some relationships in headers make my life easier? How do I decide which relationships those should be?
You clearly have opinions on how this should be done, but neither you nor the RFC explains what I gain from your way of doing things. So again, I'm not asking what I should do, I'm asking why?