I have an interesting perspective on this - or at least, *I* think it's interesting.
I am admin of a forum which is based in India, but which has a roughly equal mix of India and UK-based users. This results in some fabulously amusing language misunderstandings at times, and sadly some more complicated ones where offence is given unintentionally.
"Britisher" is used by the Indians on the board fairly frequently. (I noticed Tdol saying he hears it from Southern Asians). I have found that a lot of our British users strongly dislike it, and think it's slightly insulting. In fact, they seem to regard it almost as if it were a diminutive form.
I am sure that in the majority of cases the person using it had no such intention, but it's worth noting that it can cause problems and a pretty severe reaction from some people, so might be best avoided?
Barb, you asked if "Brit" was offensive. I wouldn't say it was offensive, but I don't think we use it ourseves very much, and it somehow feels less serious than "British".
"Brits Abroad", for example, carries some pretty negative connotations. I can't imagine ever saying "I'm proud to be a Brit"; I would choose "I'm proud to be British". Would anyone else agree with this?
Tdol, I completely agree that using English as a synonym for British is extremely offensive.