Yes there is a record- even in Indiana with an open primary., they know what ballot you took in previous elections.But there is a record of that vote and you can know for which party they voted for?
That part seems strange to me.
Can you still vote for the president and not be registered with any party for the primaries?
You can vote for any and all races in a general election (if you are otherwise eligible) if you are a registered voter- and that is not party specific.