In many areas, the words are used as synonyms. A scooter could be anything from an unpowered skateboard with a stick to a some powered wheelchairs to a moped with decent speeds, electric or gasoline. A moped is always something one sits on and has wheels, though it can include both the small and large wheels.
Even the legal definitions don't reflect that. The difference is often more dependent on the size of engine, the speeds the vehicle can go, and things like that. These distinctions can make a difference in plates and insurance.
"a vehicle typically ridden as a recreation, consisting of a footboard mounted on two wheels and a long steering handle, propelled by resting one foot on the footboard and pushing the other against the ground."
This also comes in electric versions (some of which even incorporate seats!):