The difference between Beemer and Bimmer is well known among long time BMW owner's. The BMW Club magazine (Roundel) always calls the cars Bimmers.
I've never owned a BMW motorcycle but I suspect that it's well known among long time BMW motocycle owners that Beemer is a motorcycle.
I lived in Germany for a year and had a 316I. Everyone called it a Bimmer.
Actually, I never heard cars called Beemers in the U.S. until professional people and other non-enthusiests starting buying BMWs.
It also drives me crazy when someone calls a Porsche a Porsh (even Jay Leno who should know better).