I always thought of “Briton” in that last sense, while “Brit” has the meaning of anyone living in the UK (almost). But that’s from an outsider’s perspective.
As my English cousin corrected me, though, “I’m English, ‘British’ could be anything!”. She wasn’t, of course, talking about the difference between English and Welsh, or Scots.
My first experience with this food was in Halifax decades ago. The Halifax Donair is a unique thing.
And it’s definitely Donair, not Doner.