I think it's easy to get sidetracked on "magic" vs. "law". It seems clear to me that both of these ideas are tied up in human interpretation, otherwise we wouldn't be able to have a disagreement about them, we'd simply look up the correct meaning for "magical rules that govern vampires".

I suspect that we have a fundamental disagreement that we're not going to resolve with debate, but I'll take one more shot anyway.

I appreciate that you've given a pretty succinct definition of your position: to summarize, you can only invite someone to a place where you live, although you can also invite someone into a place when you are already inside that place, regardless of whether you live there.

Can a person who lives on the street invite a vampire? If so, then a vampire is circumscribed from any outdoor location where a person lives (sans invitation); and if not, we see that "where a person lives" is not actually the deciding concept.

If you own multiple homes, which of them do you "live" in? Can a vampire enter all the others? Do you have to be in the home at the time of the invitation, or could you invite a vampire to use your summer house for a month while you're in your winter home?

All of these things cloud the idea that "living in" a place is not actually all that straightforward, and still requires the interpretation of mankind to be meaningful to the vampire. Indeed, I think the magic relies on the consent of a human, not the literal words of an invitation, and consent is innately tied to interpretation by the person consenting.

However, if anyone in the home can make the invitation, then I think the way this plays out is: the vampire cop gets a warrant, one of the other cops goes inside, and then shouts at the vampire to come inside, and then you're boned anyway.