General Discussion

When you vote in a primary, unless you vote issues only, your voting record affiliates you with the party whose ballot you voted. It is simply part of your voting record - not a separate party registration. So, if by discoverable, you mean someone can request access to the voter records, look up your name, and learn which party primaries you voted in, it is correct that it is discoverable. But it doesn't require a separate party registration - simply the act of voting in a primary.

I downloaded the voting records in my county in Ohio. I am not registered with a party (there is no such thing in Ohio). That database - free to anyone who can click on a link - includes full name, date of birth, address, date of registration, primary ballot voted for something like 10 years back, and elections I voted in for the same period. (Link here - just so you understand how easy it is to get that information). The same information is available in Texas (although I think you have to purchase it there - and I don't think it includes date of birth)

Here are the l links for all of the states, in case you want to check it out. https://www.ncsl.org/elections-and-campaigns/access-to-and-use-of-voter-registration-lists