What is fair is not objective, you're correct. However fairness does not become less important to people just because it cannot be removed from a social context, i.e. be made objective. All primates have a sense of this, we're just a bit more complex in our perception of it.

I agree with your example of the person in high school group projects, I remember that myself and it's not fair for them to be able to determine that grade. And you argue that taxes also aren't fair, and I think there's lot of truth in that.

So just because the "definition remains nebulous and largely open to interpretation" doesn't mean it's not worth trying to interpret together and trying to come to a consensus. We all do it, all the time.

Btw, the rich person would just call the cops 😅