Two bills have been introduced in the Ohio legislature to add a question about political party membership on the voter registration form. HB 208 has seven sponsors; see it here. It was introduced June 7.
The other bill is HB 210. See it here. It has eight sponsors. The chief difference between the two bills is that HB 210 has a more severe deadline for voters to join a party if they intend to vote in its upcoming primary. HB 210 was introduced June 13.
Both bills would provide for a write-in line, so that a voter could register into an unqualified party by writing it in.
Neither bill has made any headway so far. If either one passed, Ohio would switch from being an open primary state to a closed primary state.
The bills make no provision for a party to tell the state that it wants independent voters to be able to vote in its primary. In 1986 the U.S. Supreme Court ruled that if a party wants to let independents vote in its primaries, it has that right. That case was Tashjian v Republican Party of Connecticut.
Ohio independent candidates are often challenged, on the basis that they have behaved as though they are really party members. Courts that adjudicate these disputes frequently delve into which meetings a candidate has attended, and even details about their spouses. HB 208 would end that. An “independent” would be someone who is registered as an independent.