"what does it look like" and "what kind of vehicle does it share parts with" seem to be what you're focusing on. That's certainly not the only "correct" way to decide what will be called a scooter or a bicycle.

Some people focus on the legal definition in their own area to decide what's a "bicycle" or "scooter". A person could focus on what's safe to operate with very few rules (like a bicycle) and what's somewhat more dangerous to people and property (like a motorcycle) - then we might define bicycles vs motorcycles by their weight and maximum speed.

None is the perfect definition.