Yes you are quite right. There are some reasons Ohio has it's laws written this way though.
Over the years there have been bikes built by Suzuki, Yamaha, and others that look quite like
mopeds but they don't have a pedal drive system and totally rely on the motor for propulsion.
These have even been sold as "mopeds".
Ohio says those are actually small motorcycles. (Ohio reqires them to be titled, license, and insured
just as a motorcycle and the owner must have the "M" stamped on your personal operators license)
Mopeds Ohio recognizes are posted on a list in the BMV web site. If it isn't on that list, even if it was a
factory built moped, you can't license it as a "moped" in Ohio. If it is on the list, you won't be required to have
the "M" stamp on your personal operators license.
Bicycles with helper motors have to conform to the list of limitations. You won't have to have
the "M" stamp on your personal license and you DO NOT have to purchase insurance on the
motorized bicycle. But your bicycle still has to have the pedals, sprockets & chain so it can be operated by human power.
To get the "M" stamp on your personal operators license, you have to take a bike to a designated
motorcycle testing facility and take an operating test on a small track which is like a maze. If you
pass, you get the "M" stamp. On taking your regular drivers license test you will likely need to tell them
that you want to operate your motorcycle and they will set you up for the test and appointment
time and location. I have the "M" stamp on my license as I have riden motorcycles in the past.
Every 4 years when you renue your license you have to make sure to tell the ladies at the BMV
to make sure to carry the "M" over on your new license. (if they get careless and forget to
to do it, you'll have to take the test over again)
The advantage of having the "M" is that if a Cop pulls you over he will know as soon as he looks
at your license that you know what you are doing. He may have questions about your bicycle with the motor on it though. (so it may be worth carrying a xerox copy of the BMV's guidelines to show
that your bike complies with everything.
Bikes like this one are often called "mopeds" advertised as "mopeds" and sold as "mopeds" but you don't see pedals on it. In Ohio that makes it a Motorcycle, so if you're pulled over the Police will look for
the "M" stamp, and ask for regristration and proof of Insurance.