The border line has changed so many times over the years.
Ulster, according to the British, is another name for Northern Ireland, which is part of the UK. Ulster, according to the Irish, is an Irish province which isnt part of the UK. The television station in Northern Ireland is called Ulster Television, even though its based in Belfast, which is a British city rather than an Irish city, as you know.
IMO, the term 'Ulster' should only refer to the province in Ireland, and not to Northern Ireland, however a) nobody is ever going to agree on the status of Northern Ireland, and b) some of the counties within the province of Ulster are physically in Northern Ireland, which is probably why the British often refer to the whole of Northern Ireland as 'Ulster', which is technically incorrect.
Also, because there have been so many peace treaties over the years, a lot of people in the extreme north of Ireland became British citizens, and some people in Northern Ireland became Irish citizens, which is all a lot of bullsh1t anyway because both countries are part of the European Union so the individual citizenship is a little irrelevant technically.
But if you look at some British maps of Ireland, they show the island of Ireland split into four provinces - Connacht, Munster, Leinster and Ulster - with Ulster being what is essentially Northern Ireland, a term in itself which is misleading because parts of County Donegal are further north than Northern Ireland