Briton is the correct demonym for a citizen of the United Kingdom[0]. Britons is the preferred way to speak about all residents of the British Isles without having to use a qualifier, e.g. British people, People of Britain.
This is news to me and I have a problem with the term as it does not include citizens of Northern Ireland. Northern Ireland is part of The UK(Great Britain and NI), yet not part of Great Britain(England, Scotland and Wales) or Britain(England and Wales). The Irish, amongst others, do not refer to the islands as the British Isles due to the association with Britain.[1]
Yeah, there are a lot of subtleties and disagreements. To cite another example, do you include the Isle of Man? They've long been though of as a seperate people.
In addition, 'Irish' could mean a number of things (Northern or Southern/Eire, or in some parts of the world just of Irish descent).
The tumultuous political, military, and colonial history of the British Isles, Ireland, and the surrounding smaller islands has contributed to some very confusing nomenclature when talking about the places and people belonging to them.
Compare Finland, Denmark, Scotland.
[0] http://en.m.wikipedia.org/wiki/List_of_adjectival_and_demony... [1] I'm a British Citizen