There is a difference between "British" and "Great Britain". Great Britain is the island of Great Britain, i.e. England, Wales and Scotland. Great Britain is not a country, that is the United Kingdom of GB & NI. However, citizens of the UK are officially described as "British citizens". There is no such thing as a UK citizen - check the UK govt website if you're not sure. The nationality any British citizen would fill in on any official form would be "British". The term British is not restricted to those from the island of Great Britain.
People from NI can be "British citizens" or they can be "Irish citizens" - in some cases they can be both. The point I'm making is that people with a passport with "UK of GB & NI" on the front are, officially, British. Rory McIlroy, for example, is a British citizen. Obviously, and rightly so, he also considers himself Irish but his nationality is, officially, British.