I personally would not be offended at being called a 'Brit'. I wouldn't (and I think most people wouldn't) normally use it in the first person as in 'I am a Brit' unless in a very informal joking type way, more likely in the second or third . . . 'He's typical Brit' etc. Actually, I don't really know of anyone who would say 'I'm a Briton' That evokes images of a battle scarred warrior protecting his village from the invading Viking hordes :-)
Strange we don't have a single noun for this. "Brit" is common but definitely slang. We would actually say "British person" in most contexts. But in this case "I am British" is much more natural. I imagine that the Polish sentence "Jestem Brytyjczykiem" is very rarely used.