I think the ring was the main reason for the teams he listed for trades. He gave up his Bird rights, which means he can't even work out a sign and trade for a decent NBA salary. If he was going to give up his Bird rights, might as well do it for something that would bring him great joy.
Nate had never even been to the playoffs before. He is a passionate guy. If he gets his ring, I see him doing the wise move and getting the best contract he can. I'm not sure what that contract could be at this point since his time with the Celtics make him seem a far more limited player than his time with the Knicks (when he was actually playing). If we are in his market, perhaps we use part of the MLE.
I am skeptical though of his interest in backing up Rondo for a whole season. Scoring typically translates to money, and the Celtics are killing his scoring.