I think that yes, they are definitely brought up with something of their original selves in tact. That’s why they favored soldiers and other fighters over regular civilians, who usually just got the pleasure of being ghouls. It’s evidenced enough in a lot of the NPC interactions; each has a very distinct personality and is very much not a mindless slave. They simply lack free will; all of their actions are an extension of the Lich King’s will, and if they act outside his commands, they are executed.
I’m not sure what the criteria is for who gets to be one and who doesn’t, though. When they’re raising initiates in Acherus, Razuvius inspects one that seems rather dazed and doesn’t look very impressed with it, and orders its execution. There’s not a lot to indicate why exactly he wasn’t impressed— maybe they like ones that have immediate clarity of their situation? Maybe they like ones that don’t show fear? Iiiii dunno.
But yeah the difference is largely political and can be boiled down to “DKs the Lich King values” and “DKs the Lich King doesn’t give a shit about”.
Okay he didn’t really give a shit about most of them but you know what I mean.A “third generation” DK is always dead and risen- just a plain corpse imbued with some major sadism until they pick up their runeblade and become entwined with its powers. These are all pretty low-ranking, because they’re not his personal selection and didn’t give themselves over freely. I think it is largely a political difference (and not a timeline difference like a lot of people say), but I think there’s some distinction in the magic, too. But that’s because of that whole thing with Koltira’s suffering that I was talking about before and not really based on a whole lot and overall lol lore