WikiGospel
For a start the Wikipedia is not an authority on anything, just a point of reference, and secondly if you were to read through the entire article on it and not just pick from it selectively you'd see that the term 'British Isles' has no set definition, neither geographically or politically, and changes depending on who you speak to.
It is not an official term in any respect which is why it's inter-changable with things like British-Irish Isles.
The official political description of Ireland during the 19th century is that it was part of the United Kingdom, as was Britain (and Great Britain).
The official geographical description of Britain is the island containing England, Scotland and Wales. The official geographical description of Great Britain is the island of Britain plus the islands of Jersey, Guernsey, The Isle of Man etc. The official geographical description of Little Britain, Lesser Britain or Britain Minor is the region now known as Brittany (give or take).
I'm not on a high-horse about this, it's just the reality. Ireland has never been part of Britain, even when it was part of the United Kingdom.