From the Wiki link, "Open: Voters may vote in primaries of a party of their choice, the choice to be made at the voting booth. When voters do not pre-register for a party, this is called the pick-a-party primary because the voter can select which party he wishes to vote in on election day. In other open primary states, voters pre-register their party preference but the information is only used by parties for mailing lists and is non-binding.
Now, just how open is this? Let's say it's a presidential election year, and I'm interested in D for this office, R for that, D for another, R for another --- 'tain't that open.
"Blanket:" (Wiki) no longer used --- the only open primary in the list.
Lawton-Lieberman? Connecticut dems shot themselves in the foot with a guaranteed loser, in a "closed primary" --- "open," or, "blanket," the republicans and independents might have saved them the embarassment --- Lieberman is popular enough in Conn. that the republicans have let him have it essentially uncontested --- might run someone for form's sake, but not as a serious challenge.