If you live in the city (of Brighton, Hove and environs) it's meaningless to say "I'm popping into the city" as you're already there. So "town" would be the city centre, but it sounds silly saying "popping to the city centre" so common sense prevails. But the whole place is still a city...
A.S. Hi!
This may be true on paper, but does it matter in reality? Even though Lizzie's the head of state and the head of CoE, she has no effective power in either. Church and state are separate, really. Or do you have a good example that proves otherwise?