Poseidon does actually live underwater. From the Iliad:
Forthwith then he went down from the rugged mount, striding forth with swift footsteps, and the high mountains trembled and the woodland beneath the immortal feet of Poseidon as he went. Thrice he strode in his course, and with the fourth stride he reached his goal, even Aegae, where was his famous palace builded in the depths of the mere, golden and gleaming, imperishable for ever.
Hom. Il. 13.21-22
Poseidon's inclusion on the list Twelve Opympians is rather more due to him being a major deity, and a much more significant target of worship, than Hades. Poseidon was the patron god of several cities including Corinth, and held to be second only to Athena even in Athens.
Meanwhile, Hades have very few cults, and in fact, some considered Hades no more than an aspect of Zeus or Dionysus.