It is a bit confusing! 'British' is quite rarely used by actual UKers, tbh, so actually that's quite possibly something to note when writing people from the UK - places where an American would say 'British', I personally am likely to be more specific. But yeah, it's worth being a little careful about because the standard American usage of British that I've observed tends to actually mean English, often specifically southern English (which is kind of what you're using it for here), and the power imbalance in the UK is heavily tipped towards the English in quite a few ways (the south of England is by far the most economically prosperous, southern English accents are the highest prestige, the government is in London, media focuses on London, blah blah blah etc.), so the implicit erasure of the other parts of Britain is sometimes a bit grating.

Anyway! I will shut up. This graphic illustrates some other speech differences, and is kind of hilarious.