Actually... even the 2nd generation DKs are undead... including Arthas before he became the Lich King.
In Athas' case... while he didn't "die" as we would expect. His soul was sucked out of his body by Frostmorne (one of the effects the sword has on those who use it). Apparently this happens so gradually that the person weilding Frostmorne may not even realize that his body is now dead and his soul has been consumed by the sword.
As for the other 2nd Generation DKs... since their rune weapons were based upon Frostmorne, it is likely that this aspect was also included in their design. Given enough time, they two would have passed from life to undeath... just not in a noticable fashion (they weren't killed, but are still effectively dead since they no longer have a soul).
02/11/2013 11:49 AMPosted by SharliomiThe DK's we play are undead. The original ones were not.
Actually, yes they were. Undeath comes with having your soul's connection to your body severed; such was the case of Arthas and the so-called "second generation".
It's also been stated that (at least the corporeal) undead are weak to Light specifically because the magic that reanimated them creates a barrier between their body and soul. Light attempts to repair this, causing the intense, painful sensation common to all undead.
That said, the original death knights of the Scourge did not usually die to reach undeath (Falric, Marwyn, Thassarian, and Koltira are all exceptions to this as they were killed before being raised). Because of how undeath works in Warcraft, it was unnecessary to kill those who went willingly when simply sundering their bodies and souls with their vampiric runeblades would suffice.
Arthas, for example, was the first of Ner'zhul's death knights and, though he explicitly had his soul taken and kept within Frostmourne thus entering undeath, he did not physically perish until years later, when he - as the Lich King - placed the "good" parts of his soul into his physical heart before freezing and discarding it beneath Icecrown Citadel.
Edit: And I just noticed I was a bit late to that - not that it matters, hope I gave some sort of extra information on it.