Despite the name, Geoffrey of Monmouth was probably French rather than Welsh. Around 1136, he became the first person to refer to Great Britain. It was political.
Until now, Britain meant modern day Wales, with an option on Cornwall too. Living on the Welsh borderland, the Norman elite were busily trying to bring Wales under their control by whatever means necessary.
By taking the name Britain and applying it to the whole of Albion, it blurred the borders between the British Welsh and the Norman English. It also placed very firm intentions upon bringing Scotland under England's rule.
In short, Geoffrey had done exactly what the Welsh tried to do over seven centuries before, but in reverse. The Welsh thinking had been, 'This is all our land. Foreigners go home!' The Norman thinking had been, 'This should all be one land, and we will control you.'
This was all very academic and not taken seriously at the time. Which was a shame, because it transpired to be precisely what was going to happen.
In Geoffrey's vision, Great Britain would include all of the countries of old Albion - Cornwall, England, Scotland and Wales. A millennium later, it does.
It was called 'great' to differentiate it from 'little' or 'lesser' Britain. That's Brittany, in France, now. It's populated by those Celts who fled south across the channel, as refugees from the invading Anglo-Saxons.
The Celtic countries will welcome the Bretons (from Brittany) as one of us. The English barely think of them at all and, if they do, it's under the assumption that they're French.
Nevertheless, Great Britain survives as an alternative, and now official, name for Albion.
You will never get in trouble for saying Great Britain with either side. Consider it safe ground, as long as you also note one more thing. The Irish have not been part of this story. They are not, and never have been, part of Great Britain.
You will not make any friends with that mistake. You've as good as told them that they should be governed from Westminster; and a lot of blood was spilled creating that republic. In fact, I'd go as far as to say confusing Eire and Great Britain is one of the biggest errors that you could make.