There wasn't anything snobby about it. When it started, it was the only one. There was no need to call it "The British Open" because there wasn't another. It was simply "The Open Championship" because anyone (any professional, to start with) could enter.It's similar to why British postage stamps have nothing that explicitly say they're British. When they were introduced, they were the only ones in existence.
7/19/14 at 7:35pm