Lieberman is *not* running for re-election TODAY as a Democrat. Either he is a Democrat or he is not.
"Either he is a Democrat or he is not."
Well, he is not, but he is. He is legally a Democrat in the Senate until election day. Then, assuming he wins, he officially becomes an Independent who can then choose which party in Congress he wants to caucus with, which will be the Dems. He will then officially be an Independent that caucuses with the Dems, but not until the next session of Congress begins after the mid-term election is over.