He's running for the next term as an independent, but isn't he still a Democrat for the current term?
He is a Dem as long as the Dem Senate leadership says he is one. If you look at the official recordation of Senate votes, it is clear that Lieberman is still carried on the official Senate roles as a Dem. Lieberman was elected as a Dem and will remain as one until if and when he is reelected. He could then switch back to the Dem party in CT after six months.