One example: in some states, you can vote in any primary, but only one primary for each race. If there is no public record that you voted in the Reoublican primary, then you can safely go and vote in the Democrat primary as well.

I understand that, so then when you go in to vote, the first question on the screen could just be whether you want the Democratic primary or the Republican one.

And look, I fully understand how things work today, and that the parties control their primaries independently. But the fact is that it's not really possible to participate in democracy in the US without acknowledging the duopoly of the major parties, and if states can regulate them in other ways (such as only allowing to vote in one party's primary) the can certainly regulate them in other ways by requiring which party primary one voted in to be private.

What states have open primaries but don't hold primaries for all parties on the same date?

Why would holding them on the same day stop you? Are you thinking of caucuses?

In my home state of North Carolina, the primaries are "open" in the sense that independent/unaffiliated voters may ask for either a Republican primary ballot or a Democratic primary ballot when they show up. The elections are still administered by the state in the same manner as general elections, so you only get to cast one ballot. There's no need for the state to record which ballot you asked for in order for you to be limited to voting in just one of the primaries, though they currently do so.

The primary ballots usually also have at least a few non-partisan local races, and if you don't want to participate in either primary you can simply ask for the ballot that has just those few questions on it.

Good point - I'd forgotten that the primaries are all run by the states, not by parties like the caucuses. So it doesn't matter if it's on one day or not, they should be able to have a master list of "has voted" that gets marked no matter which party ballot you submitted.

So, the only scenario where you need to know which primary they voted in is for states with an open primary that can also have a run-off election, which would give you the option of voting in a D primary first round and the R primary runoff if they didn't keep track (e.g Georgia has this system - https://www.wabe.org/party-matters-in-georgias-primary-runof...)

Voting history is just that you voted. In many states, they record your party. So how you voted is pretty easy to infer. For example, the incumbent party likely won't have a presidential primary, so if you voted you're probably on the other team.

That said, for very arcane election admin reasons, if you use postal ballots, keeping your votes secret is very hard.

And if you're voting electronically, you get neither private voting or public counting.