
It’s unnecessarily pedantic to police the terminology to that degree. It’s legally called Northern Ireland. I will call it that so long as that is the legal term. It exists as its own legal entity for now. I feel that North of Ireland as a term does not acknowledge that it is currently its own legally separate territory. Should an Irish union happen (which I hope for), then the terminology may change.
Northern Ireland is the term used by both the British and Irish governments and, most crucially, by the majority of its inhabitants. I support Irish unionism but I do not support a denial of the autonomy and freedom of choice for the majority of the inhabitants of the British-controlled area. Everyone born there has an equal right to choose their destiny. I don’t care whether their ancestors arrived 200 years ago or 2,000.
Honestly I’d like to see some polling on what most people use terminology wise. A bit surprised no one has done that yet. Generally speaking though sources say most everyday people say Northern Ireland. I generally don’t care what term is used, though politically I favor unification. But I generally default to what the inhabitants call a place, with some exceptions for exceedingly recent colonial efforts (like Imbros and Tenedos)
And I have to emphasize that any Irish unification must be at the democratic will of the majority of the north’s inhabitants, and must include respect for the culture, autonomy, and identity of the Protestants. It would be unacceptable to unify only to suppress them because of when their ancestors got there. Arguments from “you deserve to live here less because my ancestors got here first” sucks and we’ve seen how that’s been weaponized in Israel.