Well, when they state an obvious falsehood in the body of an article nothing else said can be trusted.
From the Cuyahoga County Board of Elections webpage on "How Do I Declare or Change Political Party Affiliation":
Under Ohio election law, you can change your political party at a Primary Election only. This is done by requesting the ballot type for the political party with which you wish to be affiliated. Ohio Revised Code Section 3513.19, .20.
If you do not want to be affiliated with a political party in Ohio, you are considered to be an unaffiliated voter. An unaffiliated voter, which is identified by some people as an “independent voter”, may only vote an Official Questions and Issues Ballot, if there are any present, for your precinct on Primary Election Day. Unaffiliated or Independent voters can vote for candidates and issues in a General Election.
You are correct. We DO have party registration here in Ohio.