I think it's fair to say that there's some grey area between what a "bicycle" is and what a "moped" is, and that there are cases where despite having a small motor installed, the vehicle can still be called a bicycle. I think bicycles with electric hub motors added are a perfect example. They don't necessarily go that fast, and many users still pedal. I'm sure some snobs and safety obsessed people will disagree with me, but I don't think a bike with a hub motor should be banned from MUPs and bike paths.
I'm okay with the definition of what a bicycle is being restricted in sport and racing organizations. For example, prohibiting the use of installed motors on bikes in the Tour De France is certainly fair.