Actually, a lich is your definition of being undead. They are fully concious of their surroundings, can think, speak, talk, and make logical deductions. The only thing they don't do is age, eat, and other biological functions. Mannimarco was a lich in Daggerfall, and he always threw wild parties. Aesliip was a Skaal turn necromancer, who in an attempt to stave off dying too early, lichified himself in order to keep the barrier, that is blocking Solstheim from a daedric invasion, constantly active. Unless you mean they die then come back to life as if nothing had happened, with everything normal.
All a lich really is, is a person, through some rituals, has his/her soul ripped out of his body, placed in a temporary vessel, and allows his/her body to die off. Once that is complete, the soul goes back to its old, but now corpsey, body and doesn't suffer effects of aging and so on. Heck, one could argue that becoming a lich is almost like enchanting, except you are using your own soul to enchant a dead body to house you.
Either I mispoke/typed or you misread. A lich is undead, as are Zombies and Skeletons. A Vampire, however, is not, as it exhibits all functions necessary for life. A zombie is also conscious of their surrounding, as well. It is why they will attack you. Their thought process however, I could not guess.
But if you meant Lich being living (because I go by the strict definition of undead, which is a dead organism reanimated, not brought back to life or between life and death.), it is not that. They are conscious, as they contain the brain and soul of the person formerly contained. However, the body is decaying, or perhaps not. Someone more knowledgeable about Liches could better answer whether or not the body is still decaying. At any rate, a lich cannot be classified as living in the real world as it does not exhibit or function as living organisms do.
And to be classified as living (Side note: Viruses are NOT living, they're just like data that cells replicate within itself, and that is why they become devastating.) one must have the following: Able to maintain homeostasis, contain a cell (nucleus not required), metabolize energy, ability to adapt and evolve, reacts to stimuli, and can reproduce.
A vampire exhibits all those functions, save possibly having a cell and evolve (because of debates of evolution in TES, wherein some great experts and theorists, such as Lady N, I believe, say it is actually de-evolvation. ). I personally do not believe in that (laymans) theory and rather believe that things do evolve because things do undergo mutation, they're not all replications of eachother, and those mutations are passed down to the offspring. That is evolution. But here's where you can sort of rearrange some things. The virus strain can evolve, and has shown that it does. You have an unknown amount of strains, even different Sub-Cyrodiilic. The virus is now encoded in the TES version of DNA and will be passed on, by either sixual reproduction or infecting other organisms. Now, whether or not there is an equivalent of cells in TES or not, I do not know.
But yes, a Vampire does maintain homeostasis, as well as being able to metabolize an energy source (Blood), reproduce, adapt and evolve, and reacts to stimuli. Their bodies are not dead, which is required to become undead. Now, onto the next part.
It has been long debated whether or not you actually die to become a vampire. It is said you do. Maybe you do and therefore are undead. It is my personal belief that you do not actually die, but rather the virus (called a disease) simply has awoken fully. Even in vampire movies and lore, people don't die exactly. Rather, they are put in a sleeping state for a period of time to reawaken as a vampire. For some lore, it is instantaneous, suddenly after being bit. Perhaps the "On the third day, when the PC sleeps, he awakens in a mausoleum two weeks later with all of his equipment (the religion of the time being Egyptian in its belief that people should be buried with items precious to them.)" On TIL is more symbolic than literal. I mean, after all, they retain all their functions and basic components of life.
Again. People try to add RL science in a fantasy setting. The whole "Porphic Hemophilia Cyrodiilia" has a great Daedric Influence over it. Clavicus Vile is responsible for the "Drink blood, look human" sort of thing. Not strictly to logical scientific evidence.
Now... at the end of all this, it's really semantics and TES vs real world. But I think, despite how much people try to differ the two (tes vs rl), the real world can greatly be applied to TES.
Just because it may be a creation of magic, doesn't mean anything, really. I know about all that, trust me, lol. I am merely leaving those details out (which actually do not disprove what I am saying, even though I am leaving them out) to offer my opinion on the entire thing, aside from the Daedric influence, I am looking at it from a biological standpoint. I know RL cannot be applied always, but who is to say what can or can't be applied? I say it can. You say it can't. Who is right? I try to make my arguments, same as you.