Visitors have no real reason to refer to Ulster, really. Unless they're talking about Ulster Rugby being fans of the sport.
Calling it Ulster v Ulster is a local population issue, but IME not a daily occurrence. Again, that's split along political lines and not geographic ones, i.e., Republicans & Nationalists use Ulster to refer to 9 Counties whereas Loyalists & Unionists use Ulster to refer to Northern Ireland.
Some examples of Ulster being used to refer or indicate Northern Ireland only:
Northern Ireland football team are known as the Ulster Boys. "Everywhere we go, it's the Ulster Boys making all the noise!" sung by NI fans.
Great Ulster Way only goes around Northern Ireland.
Ulster Bus, Ulster Hall, Ulster Hospital, Ulster Archaeological Society, Ulster Aviation Society, etc, are all Northern Ireland specific.
But I can't see why a visitor would be using Ulster in conversation with a local irrespective of political views. Its very much a local population thing.
I sometimes call Northern Ireland "Ulster" and sometimes describe my identity as Ulster-British to an outsider.