It's a place where it is continually enforced, not responsively enforced.
It is a place where if you try to follow after someone, someone will be there to immediately respond and tell them to fuck off, not take ten minutes to get there.
They'll listen because there's a bunch of you and they're all telling you to get lost, or they'll call the cops.
And no, Urist, it's not just a relabeling, because those places have not always been open to you. That is, in fact, the point. You saying that we don't need a new term for it seems like a pointless protest, akin to saying we don't need the word church when we already have the perfectly serviceable word temple, and that therefore churches are pointless. Providing a name for something makes it a distinct thing. Sometimes those are necessary.
You aren't teaching people 'it's okay to be X in here', or 'it's not okay to do Y elsewhere'. You're telling them 'nobody is going to do Y to you for being X in this location, we've made sure of that'. Usually, those people have first-hand experience with that not being true in other places. The police cannot respond instantaneously. Court cases are expensive. Not all private spaces are willing to be a safe harbor.
You're saying that this all already applied. But if the term was created anyway, that means that that was not, in fact, the case. It's not private property inherently, it's private property because that's the only way to have a means to enforce those boundaries. It is a specific type of private property, used for a specific purpose, much like we have a word for meeting room, even though it is, yes, just another room in the office building.