This has been argued many times and it varies by the laws of each state.
The original name was a nickname for a motorized bicycle, a conjuntion of the abbreviation for Motor "MO" and pedals "PED, thus moped.
Mopeds in most states must have pedals that are functional so the rider can pedal it for propulsion.
They also must be 49 cc or less
not more than 2 hp
have automatic transmission.
Electric mopeds usually fall under a different set of laws.
In Hawaii, small scooters with 50 cc engines were called mopeds so the name stuck and was applied to small scooters and later to most all scooters.
I am a dealer so it drives me nuts when someone calls me for parts or repairs for their "moped" and I have to play Dick Tracy to figure out if it is a Goped, electric scooter thingy, real moped with pedals or a scooter.
One guy even broke down in front of my shop one day in an old pick up truck so he came to my shop and ask if I could help him. I didn't know his truck had broken down, I just saw it sitting in front of my driveway. So we walked out to his truck and I ask "so where's your moped" He look at me strangely and said " I don't have any moped, it's this truck". I fixed his truck, so I guess pickup trucks can now be called "Mopeds"
Wouldn't the world be so confusing if everyone was named "Bob" and every vehicle was called a "Moped" : )