Now, *clears throat*, as a currently-certified NYC Board of Elections Official, I'd like to answer a few points here
I think "KEEEWWLLL!!!11oneoneone" is the appropriate response to this information. And I'll break the subset of response-points out of the spoiler, being now fewer and shorter. Slightly.
I will milk my trivial position for as much as its worth, I'll have you know.
Thus you register (or re-register) for the other party. Give up your (meagre, inconsequential) influence you would have for your ideologically-matching party (doesn't need your help to go where you happily see they are going already) and instead leverage your influence (slightly less meagre and inconsequential, you might think, especially if they're vote-wise the minority party where you are) to sour the opposition's ranks.
What do you mean? Do you mean, register for the party you are not ideologically a member of? Yes, that's completely valid, but at that point you're, by definition, just a liberal Republican, which is a thing you can be.
Or do you mean re-register within the course of a single election, i.e switch from one party to another between primaries? That wouldn't work, because A) most primaries are on the same day in the same place, and B) registering can reasonably be expected to take a long time in most (?) states.
which is functionally equivalent in the "choose only one" environment.
If you are registered for a party, semi-open is essentially closed, yes. The difference is entirely for independents.
I must look up the relative rates of unallied registrations between the various varieties of Primary operation. Keeping options open would seem useful in Semi-Open, Making a choice best in Closed. In an Open situation, there'd be no advantage either way (w.r.t. the Primaries, anyway - there might be other things a registration activates or otherwise).
That would be interesting to look at.