Yes, there is or was an Ohio NY. I found it on a late 1800s map of the area. Like the Russia NY that you have for several Hinmans on your site, it was originally a part of Norway, Herkermier Co NY. In fact, it was a few miles north of the village of norway. which, was aproximatly were Captian David Hinman Settled ref: History of Herkamier Co Ny. There are several other hinmans in the area: Eleasure Jr and benjermin etc.
However, it did not become the village/ town of Ohio untill 1820s.
I am also wondering about the county of Ohio in Indiana. I remember somthing about Indiana having only three counties in the early days. Was Ohio one of them?