While skeletons, zombies, ghosts, liches and draugr all get the name "Undead", I don't think they're all the same form of being. I mean, the book Corpse Preparation describes in detail how to create the first two, while ghosts seem to be restless spirits that have no master and liches actually created themselves.
There is a huge difference because in the aforementioned book, skeletons and zombies are described as basically nothing but a couple of bones and flesh kept together by magic (the book even recommends using leather straps at the joints, so that they're more stable). They're completely devoid of life or "unlife", they're nothing but objects held together by magic. So what *should* hurt them is Dispel magic, because it removes the one thing that keeps them going.
Ghosts however, are positively dead, they're just souls in the wrong place or something like that. There are plenty of quests to free them and all that. I don't know what happens when you "kill" the aggressive ones; I guess you simply destabilize their ectoplasmatic structure, allowing them to relocate elsewhere, hopefully in Aetherius. But that's just a guess.
But then there are liches. Liches chose their fate willingly, so they must know that they won't turn into a simple assembly of bones, they're something "more". Whatever that is. They retain their ability to think and to research, and they even keep their soul, even if they have to place it in a phylactery. So they're not your standard zombie with added magical powers, they're something else. And THIS difference might just be the point where you could argue that liches (and maybe draugr) could have a weakness to Restoration, while all the other types of Undead are just not the right kind of undead for that.
I mean, turning yourself into a lich is a science in the Elder Scrolls games. You can explain it with the craziest thing you come up with, and it will make sense in a way.
Conclusion: If anything, then only liches should suffer from a weakness to Restoration magic, because they're the only Undead that are truly unnatural. The whole idea that being undead has something to do with "negative life" really only makes sense for them, and the way they're portrayed kind of suggests something like that.
Vampires are most certainly undead. Morrowind and Oblivion simply did not take it as far as Daggerfall did. In Daggerfall you very literally pass away. You're buried at a nearby cemetery, and because you died, your guild standings are all erased. MW and OB just have you die and return to life after a little while on the bed you slept in. Not quite as interesting as Daggerfall, but there's nothing there hinting at you simply having a disease that makes you want to svck blood.
There's also nothing in Daggerfall that's hinting at you being actually dead when they put you in the ground. Might just as well be a disease there, too.