It was John Dee's Idea that the English should be included in "British" to bolster Queen Elizabeth I's claims to Americas with French, as the "British" formerly had been a name for Celts in Great Britain and Little Britain (Brittany). Back then Great Britain was only the largest island, and the area was the British Isles.
Another name for Ireland was Hibernia, possibly from Latin/Romans. Wales is from Anglo-Saxon for foreigner, Gaelic also means foreigner. The Celts (Celtic) was originally a collection of many tribes from Ireland to Northern Turkey, mostly destroyed by the Romans in retribution (The Helvetii helped Hannibal. Earlier Macsen, a North Italy Celt captured Rome in retribution for Romans helping Etruscans, see story of Geese and Capitol). An agreement between Macedonian Alexander (Eskandar) the Great and Celts north of there survives.
But the Pope had earlier given Ireland to
Henry II. Ireland had never then had a national identity or been a single nation, which is why it was still not "conquered" by the time of Elizabeth I (nearly 400 years later). Elizabeth cut all international trade with Ireland and most Irish trade was then controlled by English middle men till the late 1970s (gradually aver Common Market/EU Irish-British trade dropped so low by Brexit 2016-2021).
Great Britain now includes all islands off the Great Britain mainland, but not Rathlin north of Northern Ireland, nor Rockall, which the UK annexed in maybe 1955, and as is it's not possible to inhabit it, that's contested.
The Isle of Man and Channel Islands are "British" but not in the "United Kingdom of Great Britain and Northern Ireland", which has six of the nine Ulster counties. There was supposed to be a boundary commission and there was no referdum in each county. Perhaps it should only have been 3 or 4).
From 1801 to 1922 it was the "United Kingdom of Great Britain and Ireland." Despite William of Orange taking Queen Mary's throne when his armies forced James to Abdicate (the later Jacobite wars in Ireland and Scotland were really an English Civil war), Scotland became part of the "United Kingdom of Great Britain" in 1707, so likely that's when all the Scottish isles inc unlikely Shetlands and Orkneys became part of Great Britain.
There was talk in 1922 of becoming just Great Britain when the "United Kingdom of Great Britain and Ireland" ceased, but as a sop to Unionists they put Northern Ireland.
Ireland had never been a Nation State and the Ard Rí (High King) wasn't a National ruler. So after 1922 it wanted to be a Republic. But it was still partially under British Crown, so was called The Irish Free State (Saorstát Éireann) till 1937 when it unilaterally became a Republic. The name is simply Ireland and till 1998 claimed the entire Island of Ireland.
So in the Good Friday or Anglo-Irish agreement which ended 70 years of pointless IRA terrorism, the Irish Government agreed to give up territorial claim to the 6 of 9 counties of Ulster called Northern Ireland and the UK agreed that the whole area should be called "These Islands" rather than the British Isles.
The Nation state, less 6 counties of Northern Ireland on the Island of Ireland is still just Ireland, the "Republic of Ireland" is a nickname and Éire is only when writing or talking in Gaelic, and also is both the entire island and surrounding islands inc Rathlin and the Nation State.
That is the simplfied version!
I've never in decades of living in UK and longer in Ireland heard anyone say the "British Islands", which technically would include Falklands, Bermuda, British Virgin Island etc in people's thinking.
The Chagos Is were illegally held after Mauritian independence and UN ruled on that. UK only recently agreed to give them back.