Where is that from? It's wrong on at least one fact. There is no state called the Republic of Ireland.
Britain is not a shortening of the UK, it's a part of it. The Great part of Great Britain stems from it being larger than Brittany and goes way back to Norman times when they were the one kingdom. The modern day term "Britain" then is what is referred to in "the United Kingdom of Great Britain and Northern Ireland".
Northern Ireland is a part of the United Kingdom but not part of Britain. However, the UN has recognised that when referring to persons from the UK, the term "British" can be used.
It's also worth pointing out that it is generally accepted by both sections of the community on the north that you are entitled to call yourself "British" if thats how you feel, "Irish" if that's how you feel or "Northern Irish" if that's how you feel.
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