Exactly Right

August on why the more extreme anti-Lieberman factions need to tone down their rhetoric:

If you say he's "no longer part of the party," you are not correct. Lieberman remains a member of the Democratic Party. He will continue to be a member of the Democratic Party during the remaining course of his Senate term, and if he defeats Ned Lamont, he will likely stay a Democrat. But he has not "stopped being a Democrat."...

Party nominations are not the same thing as registered affiliation. Lieberman losing the Democratic primary means he can't run on the Democratic Party ticket. Party affiliation and nomination are not the same: in New York, for example, candidates can run as the nominees for multiple parties at the same time...

And frankly, I think it's better to continue saying Lieberman is a Democrat: he's a Democrat that failed to appreciate other Democrats, attacked other Democrats, and as a result, the Democrats in his home state chose a better Democrat to represent the Democrats in the election. The talking point shouldn't be Democrats kicked Joe out of their party; it should be that Joe decided he would rather be elected by Republicans. Saying Joe isn't a Democrat makes him look like a martyr. Saying Lamont was chosen as the better of two Democrats makes Lieberman look like a loser.

Emphasis mine.