I've had cause to research this in detail (including a chat with dear old Paddy Hickey) and on this I am confident that the BBC is right and wiki is wrong. It is trivial stuff (because, rightly, Northern Irish people can choose which team they want to participate in and have done so with distinction for both) but where would pbc be without pedantry about obscure topics?
The crux of the matter is the answer to the question of whether someone from Northern Ireland who is either unable of unwilling to be an Irish citizen can nevertheless still compete for Team Ireland at the Olympics. The answer to that question is that no they can't. Therefore de jure Team Ireland is the team for the Republic of Ireland, not the land mass of Ireland.
People born in Northern Ireland are not automatically Irish citizens. It is a right that needs to be exercised, and they have an option never to exercise it. It is also theoretically possible for someone to be born in Northern Ireland, to be a British citizen, and to be ineligible for Irish citizenship, depending on who their parents were.
The Olympic Council of Ireland may well have text written into it's constitution claiming that it "represents" everyone on the island of Ireland and such, probably written before the Republic of Ireland abandoned it's claim on Northern Ireland for an aspiration in 1999, but if a non-Irish citizen from Northern Ireland cannot compete for the Irish Olympic Team then it is the team for a state rather than a land mass, and that is the IOC position. The borders of that state are recognised by the UN and every country in the world as consisting of 26 counties, not 32.
The crux of the matter is the answer to the question of whether someone from Northern Ireland who is either unable of unwilling to be an Irish citizen can nevertheless still compete for Team Ireland at the Olympics. The answer to that question is that no they can't. Therefore de jure Team Ireland is the team for the Republic of Ireland, not the land mass of Ireland.
Also note that this is not a general truth for "Ireland" teams in international sport. For example anyone born in Northern Ireland can play for the Ireland Rugby Union team, even if they have neither British nor Irish citizenship.
I wrote to the International Olympic Committee several years ago to ask why there is a GB team in the Olympics but not a UK one. The answer was basically that it used to be "Great Britain and Empire" (which included Ireland anyway) so the GB name continued after the Republic of Ireland split away. Athletes from Northern Ireland simply choose whether they prefer to compete as part of the GB team or the Ireland team.
The actual official name of the Olympic team is "Great Britain and Northern Ireland". "Team GB" is just a brand name used for marketing. You are correct to say that there is no UK team in the sense that the official name is actually "Great Britain and Northern Ireland" rather than "The United Kingdom of Great Britain and Northern Ireland" and also in the sense that the team includes places outside the UK such as the Isle of Man and the Falklands Islands, however there is actually no team called "Great Britain" only a team called "Great Britain and Northern Ireland" which has a marketing name "Team GB". If you see "Great Britain" on a medal table it's actually technically incorrect, it should either be "Great Britain and Northern Ireland" or "Team GB".
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People born in Northern Ireland are not automatically Irish citizens. It is a right that needs to be exercised, and they have an option never to exercise it. It is also theoretically possible for someone to be born in Northern Ireland, to be a British citizen, and to be ineligible for Irish citizenship, depending on who their parents were.
The Olympic Council of Ireland may well have text written into it's constitution claiming that it "represents" everyone on the island of Ireland and such, probably written before the Republic of Ireland abandoned it's claim on Northern Ireland for an aspiration in 1999, but if a non-Irish citizen from Northern Ireland cannot compete for the Irish Olympic Team then it is the team for a state rather than a land mass, and that is the IOC position. The borders of that state are recognised by the UN and every country in the world as consisting of 26 counties, not 32.