A category is just the set of (member, category) pairs. If it's empty, it disappears. It can't have a parent or anything. But it can exist (ie have members) without a page.
So. I agree that this is the formal, logical, definition of a category (in formal, logical, mathematical contexts). I'll point out that no one else is talking about categories in this way, here, or in previous discussions. But, I'll try to parse the point you're making. Please, tell me if I misunderstand or misrepresent you...
If category A is a member (ie a so-called "subcategory") of category B, the members of A aren't members of B at all.
I think what you're trying to say is that, everyone who is in the category of "Richmond, Virginia" automatically belongs to the category of "Virginia". And because Bill is also in the category of "Dunromin Cemetery, Richmond, Virginia" he's automatically in the category "Richmond, Virginia", as well as "Virginia".
Yes. Yes, this totally makes sense.
And (again correct me if I'm mischaracterizing your argument), you take issue with the fact that we're saying that "we should put Bill in both the cemetery and the city category" because that's inconsistent, and unintelligible, from the strict formal definition of categories (after all, he's already in the city category). You, I think, are effectively saying that "because you're using categories in an arbitrary and illogical way, any structure you claim to have is gibberish."
So, yes, I agree. I'd say that sometimes the official project guidelines about how profiles should be displayed on category pages departs from the way they'd be displayed if we were using the word 'category' in the formal logical sense.
But, for me at least, it seems transparent that most people are using the word 'category' in a different way here. By this, they mean "a collection of category pages, organised roughly around a pragmatic sense of formal categories", and the rules about which profiles should be displayed on which category pages are pragmatically defined as "whatever avoids cluttering too many upper level categories, and also makes it easy to find a profile by descending, in a hierarchical order, down a category tree".
I'm trying to understand how there is anything wrong with this usage. And why the formal usage of category you're insisting on has anything to do with personal vs project vs private categories.