What makes a Lich "Undead"

I’ve read both Tomb of Horrors and Tomb of Annihilation (and played ToA) and I much prefer Annihilation.

Tomb of horror is a death trap, you are comparing a dungoen that is a death trap to a dungeon that is an actual lair !
there is a clear difference between death traps and lairs. if you dont like death traps then of course the dungeon for tomb of horror will seem badly done.

from the 5e book there are numerous different dungeon types..

Death traps = made to kill players. aka prison like or maze like with no real way of escaping.

Mazes = generally to hide something or lose people you don't want to see.

Strongholds = castles or the likes, places that are fortified to protect someone of importance.

Plana Gate = a place of great importance where deities or their minions can be summoned or conjured.

Vault = good old fashion tresure room, well protected and usually well guarded.

all of these are very different dungeons.
you can't just compare each of them against the others, as they are very different in their motivations.
tomb of annihilation isn'T a death trap, its literally a lair, a stronghold.
what would be a death trap is the bbeg capturing a player and then sending him into a room designed to just kill him.

tomb of horrors is also a mega dungeon... its a campaign in a dungeon all by hitself.
imagine if tomb of annihilation was hapenning inside the last dungeon at all times, no jungles no cities, nothing... you'd have had a very different experience out of it.

as for liches.... their life force starts to wane if they don't kill people on a daily basis. since people tend to avoid liches and their goals, it is normal to sdee liches decrepit over long period of times. but thats why acererak decided to create the syphon which kills people of the world and bring their life force, soul, to him. that way he get a sustainable source reguardless of where he is.

All those are reasons why I liked Tomb of Annihilation so much better than Tomb of Horrors.

Quote from Golaryn >>

D&D Lore is a muddled mess of inconsistencies and retcons.  It is best to define the lore of your game world yourself instead of sifting through all the various editions and trying to piece it all together.

 I just never moved on from 2e. I just kept updating things to the new rules (skipped 4e though), but kept the rest the same. The rules we played by changed from edition to edition, but the setting and cosmology I use didn’t. Keeps it much simpler this way. Pick whichever you like and use that. The rest falls under the “pirate code adjacent” category for me, more “guidelines.”

Yes they are alive, but they are ALIVE, not Undead, a distinct difference is that they did not die in the first place. A vampire or ghoul or any other undead requires an initial death, whether the body is animated through magic or the souls "return" or some other means usually means the body (or ectoplasmic representation of a body like a ghost has) is undead. 

Resurrection is the other option as that's the same as a soul returning, but the body is operated by normal bodily function with the soul at the helm.

A warforged was constructed, like a flesh golem, that's why they are constructs, and they did not perish and shuffle off the mortal coil, that's why they are not undead

Resurrection is what you have done to this over 5 year old thread.  8)

Warforged are not Undead for the same reason Flesh Golems are not Undead.
A Flesh golem is a large being made by taking the parts of various humanoids and stitching them together and animating it, which so far would qualify for undead. But. The method used for creating a flesh golem is by using a book to read and put together the parts, then burn the book and sprinkle it. This means that the flesh golem was designed and built by other creatures, not a spirit or chaos magic surge.
Warforged do not decay, but they do not breathe eat, sleep, or drink, which is the same as various undead. However, the same can also be said of a clay golem, an animated armor, a rug of smothering, etc. Warforged souls are created by using creation forges that can only be used by House Cannith dragonmarked individuals, which makes them the only construct with a soul. If you were to reclassify Warforged as Undead because it has the aforementioned similarities to undead would require reclassifying all constructs. The premises for this creature change notwithstanding, Warforged must remain the same.

As for Liches being Undead, you could decide that they are not Undead, but what other creature type would you give it? They are not Aberrations, Beasts, certainly not Celestial or Dragon, if you make them Construct then all Undead could be shifted to Construct for various reasons, Fey, I suppose that Fiend could work similarly to a Rakshasa. . . Liches are Undead, and it would be hard to find a reason to move them to any other creature type.