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Those are actually two entirely separate things. Membership in a local political party in the US is unrelated to whether you vote for that party. Party members are more active in organizing and generally have to be approved by other party members, whereas anyone can register as a party voter (in states that require/allow that).
Oh, it gets even worse. Some states require party identification when registering to vote, some are optional, and others don’t ask about it at all. Relatedly, the party primaries can be “open” (anyone can vote in it) or “closed” (only party-identified voters can participate). Also, the party may choose to do a caucus instead of a primary, so instead of just casting a ballot to choose a nominee, registered party voters select party members who have declared for a nominee and then they hold votes to see if a nominee has enough votes to win.
And that’s the simplified version. It’s a total clusterfuck and if there was a national standard that was easily understandable, things would be so much better.