[denethorthefirst]

Back to topic: why Sauron wants the Ring?
Simple reason that applies in both cases: he wants it back, because its his! The ring is a part of him, thats reason enough for wanting it back!

If definition 1 is correct:
he wants to recapture the Ring simply to deny any possible contender access to it. Recapture of the Ring would also speed up the conquest of Middle-earth by a few months or years because the works of the three elven rings would be "laid bare" (but given the fact that Sauron is hundreds of thousands of years old, i dont think that a few years here or there matter that much to him).
But Sauron doesnt seem to "need" the Ring that much, he can conquer Middle-earth without it.

If definition 2 is correct:
he wants to regain the Ring because it would make him much more powerful; he really has to gain something by recapturing it.