The idea is that which person/party is going to vote for would matter in a general election because if you’re a say a tricky fraudulent Dem/Hillary supporter, you don’t want to bother committing fraud by voting as someone who would have voted for Dems/Hillary anyway.
It the situation we were talking about, person B although registered wasn’t going to vote.
The primary fraud idea is somewhat silly - if you end up stuck with a ballot on the other side, what’s the point really?
Well, you could vote for someone you thought had no chance against your own party’s candidate in the general election.