Being Northern Irish myself I'm quite aware of that. It's specificially why I pointed out the Team GB discrepancy. British is a nationality, it doesn't mean "from Great Britain".Except that in most of the contexts you quote above the term British is being used incorrectly. So for example the Olympic team is actually the United Kingdom of Great Britain and Northern Ireland. It is incorrectly referred to and shortened as GBR rather than UK.
Great Britain does not include Northern Ireland.
The misuse of the term is a source of constant annoyance to many in Northern Ireland.
The misuse of the term Union jack is nothing at all to do with QI. The flag became known as the union 'Jack' when it was flown on the Jack mast of a ship (a small mast mounted on the bowsprit of the ship) Flags flown from this mast were traditionally smaller than other flags. When the union flag (or as it was called then the british flag) came into use in the 1600's it was ordered to be flown at the front of all ships, so it became the 'jack flag'. The term union jack and union flag are pretty interchangeable, however the flag is officially known as The Union Flag.
Again, I meant the popularisation of the incorrect idea that the flag is officially called the Union Flag and not the Union Jack. You're correct about where the name comes from, and it is normally a flag, not a jack, but a name is not a description and it's called both. This is about the most official source on the matter: http://www.flaginstitute.org/wp/british-flags/the-union-jack-or-the-union-flag/
By 'union' it explicitly refers to the union of Great Britain and Northern Ireland. It is NOT the flag of great britain alone - although again that is often how it is often used. The term union is specific to the union of Great Britain With Ireland, which since 1922 is the union of Great Britain and Northern Ireland, the remainder of ireland returning to its previous independent sovereign nation status. The term union in United kingdom is NOT a reference to the union of England, Wales and Scotland.
Again... Northern Irish. Yes, United Kingdom of Great Britain and Northern Ireland. The nationality is still "British".