Yes it IS! Second largest island in the British Isles:- https://en.wikipedia.org/wiki/Terminology_of_the_British_Isles#Ireland
Absolutely right.
The island of Ireland is in the British Isles, being the second largest island. The largest island is, of course, Britain/Great Britain. Britain and Ireland are two distinct island in the British Isles, therefore no part of the island of Ireland can be part of Britain/Great Britain and vice versa.
So Northern Ireland is part of the United Kingdom (of Great Britain and Northern Ireland - to give it its full title).
Northern Ireland is not part of Britain/Great Britain (as that is a separate island)
Northern Ireland is part of the British Isles (being the collective term for a group of islands, the largest being Great Britain, the second largest being Ireland).
