[RPG] What are the best computer generators for historical timelines?

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.