Here in OH we just tell them which ballot we want, D, or R. If you say D you can only vote on D primary candidates, same for R.
My county is pretty much R down the line, so the R primary winner for county office is pretty much the winner of the general by default. So the D in my county have to decide if they want a vote in the county offices or they want to vote for state and national D candidates in the Primary.
So perhaps that makes us an "open primary" ??