Voting in primaries

While I have generally voted on everything from if a restaurant should get a liquor license to who should be the county coroner I have never voted in a primary. One the one hand I care greatly about privacy so I certainly don't want to publicly register an party affiliation. (Now that I am registered in Massachusetts I don't actually know if it is known which primary I voted in, at least beyond my Town Clerk's office.) On the other hand, one might reasonably deem me to actually be an independent and, perhaps through having grown up in Britain where the choice of leader is an internal party matter, it feels wrong to me to meddle with the affairs of a party with whom I feel little affiliation.

I rather like the idea of approval voting (vote for as many candidates as one likes) as being a good balance between comprehensibility and expressiveness and immunity from dysfunctions like Arrow's theorem. Could we so vote then I might be tempted to at least turn up and vote for all the candidates who actually seem credible to me, but we can't.