Ireland was controlled by Britain in the 17th century (I hadn’t realised Ussher lived that long ago), but it was still officially the Kingdom of Ireland – it wasn’t combined with the Kingdom of Great Britain until 1801. Yes, the king of Ireland and the king of Great Britain were the same person*, but I wouldn’t include the Irish as British any more than I would call the inhabitants of any of the colonies British.
Just as today the queens of Australia, Canada, New Zealand, the UK, and twelve other countries are the same person.
I wonder if she has a separate crown for each of those…