You seem to have found all the contenders. The only undead in 5e which are intelligent, independent, and you can deliberately become are liches and mummy lords. To be a vampire requires another vampire, and you would then be subservient to the original vampire anyway. There are other kinds of powerful undead (such as wights or revenants) but none that you can deliberately and purposefully become.
Note that it's impossible for a single mage to become a vampire or free-willed vampire spawn, because death will end their spells, and the vampire they were trying to use to become a vampire will control them. However, with the help of another mage, it becomes very simple. Mage A casts Dominate Monster on a vampire, then forces the vampire to drain Mage B's blood and then give Mage B some of the vampire's own blood. If Mage A also wanted to become a vampire, Mage B could make Mage A one easily. (Although that requires a lot of trust between them, because Mage B has the chance of having complete control over Mage A, which most powerful evil mages would jump at.)
Lichdom is the usual method of immortality for powerful evil mages. For a bit of variety, some of them could have let themselves go, (as it were), and become demiliches. Likewise, some could have deliberately become demiliches as described in the sidebar on Acererak. A really clever mage could even have transformed themself into a dracolich, with a bit of work and some help from some of his friends. (True Polymorph plus the ritual to turn a dragon into a dracolich.)
However, it's your campaign, and you are free to do whatever you want. If you don't want your mages to be liches or mummy lords, they don't have to be. You can change the lore for an existing undead, or just create your own to suit your needs.
This is a matter of story and should be treated like one.
The answer here is not to dig through books to find something like this. It’s to discuss your character’s goals withe the GM and state that your character if always on the lookout for things that might be able to grant this boon. Make the quest to get it part of the game. There could be long-lost relics, gods willing to offer it for completing some quests, or other options. Just like you wouldn’t look to find how to get revenge on the man who slaughtered your village in a sourcebook, you shouldn’t expect a sourcebook to answer this dilemma.
The problems with digging through books are twofold.
You probably won't find exactly what you’re looking for, you’ll probably find immortality, which means now the elf has the same problem you currently have. In 3.5 there actually was a prestige class that gave elf-like longevity to other races (ruathar, Races of the Wild), but that is not in Pathfinder. You could try to reincarnate into an elf, but that is really awkward—and not necessarily what your character really wants.
This is more systematic—the game’s rules only have one way to make something special: they have to make you pay for it. Immortality should be special (or else everyone would have it), so the game makes you pay quite a lot for it. But there is a mismatch between what you’re paying (metagame character-building resources, such as feats, levels, or whatever) and what you are getting (in-character narrative benefits, which in this case have zero metagame value).
The GM can handle this situation vastly better than a rulebook can, because the GM controls the entire world—and thus can require you to do things in-character, via the narrative, in order to get those narrative benefits. And he can and should do so without making you expend metagame character-building resources for it, because it has no metagame character-building value. It’s story, not character building.
This problem is actually remarkably similar to the problems with the Leadership feat (though that is definitely not at risk of being wasted resources; in that case, it is often overpowered). Leadership should be built up through roleplaying and narrative; you should have a cohort and followers because you are someone people want to follow and so on, not just because you took a feat. Likewise, you should have something rare and special like immortality because of some grand quest, not because you took a feat, mythic power, or class feature.
Because that kind of mechanistic approach just isn’t satisfying unless it gets worked into the story anyway. “I hit \$X\$th level, now I have immortality babe, we can spend eternity together now!” just kind of falls flat. Obviously you could (and should) do more to make it part of the story, but if you are doing that anyway, why bother digging through books?
Best Answer
(Heavily adapted from my answer here)
High level spells
The easiest way to achieve effective immortality is to use spells. A wizard has access to several of them once they get higher level spell slots.
Sequester will allow the PC to put themselves into a state of suspended animation during which they will not age. This will allow them to span any amount of time without aging (though it will mean that they are out of action for that time).
Clone will provide the PC with fresh young bodies to use once their older body dies. With enough of these, they can last an infinite amount of time. See this Q&A for more discussion on this.
Wish obviously wish can do anything, including granting immortality. A wish for immortality is unlikely to be granted in a straightforward manner though and would be a good opportunity for the DM to implement some sort of curse or other downside on the PC as a result of their wish.
Imprisonment would work in any of its flavors since the targeted creature does not age while under its effects.
True polymorph can allow you to extend your life by changing into a creature with a longer lifespan than you. Some good potential options: devils are immortal (MM), hags are "virtually immortal", and ancient dragons are listed as living for "over a thousand years".
If your PC isn't a spellcaster you can always try to find them assistance from someone or something capable of casting the spells. For example, Djinni can cast wish as well as some magic items.
Magic items
(Tomb of Annihilation spoilers)
Non-spell/item options
High level Druid or Paladin
At 18th level, druids get the Timeless Body feature:
Combine this with a long-lived race like the elf or warforged and you easily have thousands of years of life.
An Oath of Ancients paladin at level 15 gets:
Which seems likely to be able to prevent death from old age as well if your DM interprets death from old age as a drawback.
Warlocks with the Undying Patron also have a similar ability. (thanks @Himitsu_no_Yami)
Epic Boon
A DM can grant the Epic Boon of Immortality to give a PC immortality. This assumes that they have a 20th level PC.
Long-lived races
Races like elves and warforged can live for an incredible amount of time (unknown even in the case of the warforged).
Starting as one of these races can get you a good way towards immortality without anything else, and will enhance the effects of many of the options above.