I live in Ohio. As I understand it a motorized bicycle or motor assistance bicycle (either way) is classified as a moped as long as its less than 50cc, has a max speed of 20MPH, and has no more than 1 brake horsepower. Pretty strict regulations. So you have to meet certain equipment requirements (turn signals, brake light, 100ft horn, etc.). Anything over 50cc, 1 brake HP, or 20MPH is legally considered a motorcycle in Ohio so you'd have to have your motorcycle permit or license. Then you have to meet stricter equipment standards, but not much stricter than a moped. As someone said earlier, you could probably register you're moped motorized bike as a Wildfire. My dad owns a Wildfire "Truck" tricky finding someone to insure them. As for a motorcycle motorized bike, well if you have a Honda or Robin-Subaru engine on it you could probably register it under the engine model. As for the popular generic Chinese two-strokes, I'm not quite sure how that could work. Possibly name it after the company you bought the engine from such as Golden Eagle 80cc.
I'm not quite clear as to the "meeting the list of qualified vehicles" I see no reason why it could not be added to the list as long as you meet all their listed equipment requirements. If you wanna suck up you could get one of those catalytic converter mufflers and add a little bit of alcohol to the gas (just for the test, none after) and get an emissions test and pass it. Moped manufacturers and motorcycle manufacturers must have to have their vehicle added to the list, and I see no legal reason why a large corporation could get an approval but a normal citizen couldn't. Then the trick is finding an insurance company to insure it. My dad found one to insure the Wildfire "Truck" and that thing looks way more awkward than a motorized bicycle, so I'm sure some insurance company will insure your bike.
I'm in the planning phase to build my own "motorcycle" motorized bicycle. I haven't had to opportunity to ride one yet. Once I get mine built I plan on doing everything possible to get it registered and street legal. There are all kinds of loopholes in the law. Even if I can't get it registered, I'll still ride it. Hit the kill switch when I see a cop, and flip a secret kill switch under the seat. Have the officer flip the visible handle bar killswitch and try to start it and it does nothing. "Sorry officer, I just thought it'd look cool having an engine mounted on my bike even though it doesn't run. Sorry to waste your time."
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I forgot about the motor themselves. I believe for on-road use the motor would have to be EPA and/or (not sure which ) C.A.R.B approved. I'd assume that leaves most of the generic two-strokes out, but I'm not sure. I believe the Honda GX35 is approved though, so thats what I plan to use.