By this definition, Northern Ireland is part of Ireland and the inhabitants are still called Irish. I don't know a ton of Irish people but I think I've heard them say it like that too? On the other hand, there seem to also be contexts where "Ireland" is shorthand for "Republic of Ireland".
In the UK, "Ireland" can mean either the country or the island as a whole. It's often clear from the context which is meant, though "Republic of Ireland"/"Irish Republic" is frequently used instead to make the distinction obvious.
The Good Friday Agreement[3] recognises that the people of Northern Ireland may consider themselves British, Irish, or both:
> recognise the birthright of all the people of Northern Ireland to identify themselves and be accepted as Irish or British, or both, as they may so choose, and accordingly confirm that their right to hold both British and Irish citizenship is accepted by both Governments
[1] Constitution of Ireland, Article 4: "The name of the State is Éire, or, in the English language, Ireland." https://www.irishstatutebook.ie/eli/cons/en/html#part1
[2] The Republic of Ireland Act, 1948, section 2: "It is hereby declared that the description of the State shall be the Republic of Ireland." https://www.irishstatutebook.ie/eli/1948/act/22/section/2/en...
[3] https://peacemaker.un.org/sites/peacemaker.un.org/files/IE%2...
One of these days I’ll refer to “Southern Scotland” in return.
Perhaps you could use 'Southern Albion' instead? Or 'the south part of the Eastern Island', maybe.
Even Wikipedia erroneously catalogues "Ireland" under "Republic of Ireland".
However! That page makes it clear that the name of the country is Ireland:
Ireland (Irish: Éire [ˈeːɾʲə] (listen)), also known as the Republic of Ireland (Poblacht na hÉireann),
since Ireland there takes precedence.
In conclusion, I think it's just a technical limitation.
Some other posters have replied with some good information, but if you want Entirely Too Much Information this wikipedia page has a more fleshed-out discussion.
https://en.wikipedia.org/wiki/Names_of_the_Irish_state
Even that is missing some more exotic parts of that history - I see no mention of 'Republic in Ireland', for instance.
Variations of The Republic are used for disambiguation but are not used officially.
> Northern Ireland is part of Ireland
The island yes. But not the Republic of the same name.
> and the inhabitants are still called Irish.
A large portion of the inhabitants see themselves as Irish. But there is also a large portion who see themselves as British. These are not necessarily exclusive categories as some of them see themselves as both British and Irish. There are also those who see themselves as Northern Irish.
Full disclaimer: Not Irish or British, just have an unhealthy interest in the situation.