In the beginning, kings had it magically made so that any of them could travel there in a day or something like that, and put magical protections on it to prevent their using it for anything but its purpose, namely negotiations.
Grew from there. Royalty went there to show off their eligible sons and daughters and otherwise associate with equals. Nobility followed for all the usual reasons.
It was also that go-to place if you were overthrown, since, after all, you could not be winkled out of it. You could even raise troops there as long as you didn't fight with them, or intend to conquer.
and now I think the outside world is suffering from revolutions, which means more deposed nobles and royalty than ever.
Now, every kingdom would, no doubt, have a name for it. The question is whether it would be the official one, or just an exonym. The settled locals might just call it the city -- after all, you don't want to offend a customer by calling it by a different name than he's used to. OTOH, it's quite likely that a lot have the same name for it. Kings City, or Peaceable City, or something. . . .