You make an incorrect assumption that the people of Britain are British, regardless of ethnicity. Many Welsh, Scottish/Scots/Scotch and English people do NOT identify as British. You also missed the Cornish, and the Manx but they are on a totally different island from either. British technically refers to someone or something from the island of Britain, but is has also come to be a euphemism for England which people that are not English resent. The English can resent the term for perceived political correctness or due to national and ethnic pride.
Anyways, things from the island of Ireland can be called Irish, or Gaelic, which is the ethnic subgroup the Irish are in, or Goedilic (which is the same thing but more academic). If they identify with Britain or British heritage (and another name for Northern Irish is Ulster-Scots, because there were a large number of Protestant Scots transplanted to Ulster, northern Ireland) they could also be called British, although less technically accurately.
There is a pretty long list that you can call any one of these groups, everyone but the English are Celtic. The Welsh and Cornish are Brythonic. The English also get called Anglo-Saxons or Saxons (or Sassanach, which is Scottish for Saxon and a swear) or WASPs (white-Anglo-Saxon Protestant, basically means super white).
As for changes over history yes. The acceptable terms have changed with British political de-evolution (as it is called) as Ireland acquired it’s independence, Scotland and Wales acquired home rule and local Parliaments. In the 18th century, around the time of Union, there was a move to re-lable Scottish people “Northern English” (which is a bit different from Northerners from the North of England). English was used interchangeably with British to refer to anyone from Ireland or Britain (or the other associated islands). In the medieval period English writers often refer to Gaelic people as being Irish, even if they were Scottish. Shakespeare does this in MacBeth, so even well after the medieval period. There is also a complicated history of Viking and French immigrants in parts of Ireland and Britain, as well as Normano-Anglo (French people that were largely aristocratic) in Ireland. Arthur Wellesley, the Duke of Wellington, was from Ireland, but his family and culture was English and can be traced back to the Normans, who restricted the use of English in England for 250 years.