I agree with that but it is not true in Ohio. They can vote in any precinct within the county on the provisional ballot.
Are you saying that I cannot change my mind in a primary--as to political declaration.
it's important to remember that the registration helps the get-out-the-vote effort because of the phone numbers, mailing addresses (for lists, etc.), so the respective parties want that info to show up on their precinct lists under their party name.