"Briton" is generally used as the noun form of "British", so when "Brit" is used as a noun - which is most of the time - it's abbreviating "Briton".
As for who gets to be called "Briton": In the loosest sense, anyone with residence in Britain can be counted as British when they're here, whether or not they're considered ethnically British (by themselves or others).
Bear in mind that "Briton" originally mean "an inhabitant of the British Isles before any of the Romans, or various flavours of Germanics turned up". There's been quite a bit of admixture since then. It makes sense - to the chagrin of the Welsh, no doubt - that the term has mutated a bit over the centuries.