Ok first things first, a lot of people are getting in a huff that the OP said Ireland is in Britain: He didn't, he said 'N.Ireland'!
Yes this whole thing is annoyingly confusing, and the Scots, Welsh and Irish seem to make a point of being annoyingly picky about it! But as long as you never use 'England' or 'English' to mean 'Britain' or 'British' you should get by!
I've never heard that distinction between Britain and Great Britain, I don't see that there's any point to it. All you really need to know is that GB means England, Wales and Scotland and UK adds Northern Ireland to that. And (the rest of) the Irish are entirely seperate from all o' that there business!