It's a little silly, really. As a half-American-half-Englishman myself, I think it's fair to say Americans pretty much only do it for the sake of clarity. If they ran around calling the British Open just "The Open," people wouldn't immediately realize what they're referring to, so it would all get pretty confusing pretty quickly.
Either way, it remains an ongoing debate that will probably never get resolved. Or will it?
Scotland's recent push -- voters go to the polls next week -- to become a fully independent nation could end it once and for all. If Scottish voters vote "yes," they're choosing to no longer remain a part of Britain or the United Kingdom.
For now, the British Open is technically correct, but if Scotland, with all its rota courses, opts to go it alone, it'll mean that most of the time the "British Open" would actually be held outside Britain. Logically, it really wouldn't make any sort of sense at all -- calling it the European Open would make more sense than calling it the British Open.
At that point, it's safe to say the Open Championship would win the day.