Follow the Moped laws in Ohio. Motored bikes are treated as mopeds there.
Here's a site with some of the Motorized bicycle codes of Ohio.
I don't have the time to peruse it all, but if you're legal, it might be a good idea to print it out and carry it with you as you ride. Sounds like the po po is trying to intimidate you into leaving the streets without knowing for sure if you are breaking the law. They used to do that stuff here in South Carolina back in the '70s. Now they are friendly to motorized bicycles.