Administrative boundaries in Ohio are represented by administrative boundary relations.
For reference, the boundary relation for the United States is 148838, and the boundary relation for Ohio is 162061 . The Ohio boundary relation extends into Lake Erie to meet the Canadian border. It also extends partway into the Ohio River but excludes any river islands.
County boundaries were imported from USGS data, which is accurate but has lower resolution than other types of data in OSM. Municipal boundaries were imported from TIGER 2005 shapefile data, which is often very high resolution but contains countless inaccuracies. We've cleaned up many of the problems, but much work remains. In particular, the TIGER import included CDPs but not townships.