The nazgul were
not dead - as they have
never died. "Undead" describes them well. Their bodies had faded, passed into the Spirit world, had become invisible, but they were still MATERIAL. The nazgul could wield swords, ride horses and interact with the Physical world. If anything, they resembled faded Elves, the Lingerers, but their origins were, of course, different.
The victims of a Morgul-wound also fall into the same category, undead wraiths, much like the nazgul, but weaker and under their command. Frodo was fading when brought to Rivendell, becoming a wraith, but he was not dying. Reading the description of Minas Morgul attentively, one might assume that there were quite a number of such wraiths gathered there.
The Dead of Dunharrow, in contrast, were indeed
DEAD, not "undead": their bodies had long turned to dust, but their fëar were held in ME by their Oath. They were simple naked spirits devoid of material bodies - ghosts. They could only interact with the living by the fear they caused, but they couldn't wield material swords or ride normal horses. I believe they were entirely powerless against the fearless.
There were also Houseless Elves - In a way they resembled the Dead of Dunharrow. Their bodies had been slain and long gone, but their fëar refused the summons of Mandos and remained wandering bodiless in ME, seeking new bodies to inhabit. Thus they were quite dangerous to the Living.
Some scholars like Alvin Eriol (
http://www.sf-fandom.com/vbulletin/s...ad.php?t=20554) speculate rather convincingly that the Barrow-Wights were Houseless Elves housed in the decaying bodies or skeletons of the long-dead Dunedain buried in the Barrows.