tl;dr: Demiliches were originally weaker than regular Liches back in AD&D.
The Lich is found in the 1977 Monster Manual, and the Demi-lich is found in the 1983 Monster Manual II. The Demilich's description is a duplicate of that found in the Tomb of Horrors adventure. Back then, a Lich was what we might think of today as a template. It had the powers of an 18th level Magic-User or Cleric, in addition to being quite frightening. Demiliches on the other hand, were an entirely separate creature - the Lich's soul had long since departed, "and the evil soul roams strange planes not known to even the wisest sages." This Demilich is explicitly described as being unable to harm, only to threaten, though players attempting to defeat it will find it grows more powerful with their attacks and might manifest a Ghost or a Wraith, but never the full power of a Lich. That said, the Demilich was quite dangerous if touched - you died, instantly and without a save, and your soul was trapped. The Demilich is basically unchanged for AD&D2.
When they redesigned the Tomb of Horrors for 3rd edition in 2005, they had a problem on their hands - they'd made Demilichs into all-powerful beings of epic power when they released the Epic Level Handbook in 2002 - certainly not the same sort of being in the original module. These Demiliches are templates that go on top of the Lich template, and are described as being an advanced form of lichdom. So the Demilich in the 3.5 version of Tomb of Horrors adventure is described as a Demilich-like construct instead - to match the power of the original.
I can't speak to 4th edition, but I'm not familiar with any reference to Liches needing to consume souls to avoid becoming a Demilich in 3rd edition or prior. This seems to be a new invention of 5th edition. However, I like this new lore, as it adds to that from AD&D - the original Demi-liches were created when a Lich's "undead lifeforce begins to wane."
As for the use of the word Demi which would typically mean half or lesser, the original version was lesser in that its mortal presence had been diminished significantly, so demi makes sense in the context of the original creature. They do go out of their way to say that its only the Lich's body that is demi, and that its soul is off doing cool stuff you pathetic mortals couldn't even dream of.
Your party wizard is safe, probably. A wizard in your game might wake up one morning as a Nothic. It's not for no reason though, it's because of Vecna's dark curse. They were most likely trying to uncover some dark secret, and got in over their head. Ultimately though, your game may not even have a deity named Vecna, so it's up to you to decide what a Nothic really is.
The answer to "what is Vecna's curse" is entirely DM dependent. The entries in the monster manual aren't necessarily intended to form a complete world, but to provide enough inspiration to the DM to be able to incorporate the monster into their own world. Vecna is a powerful lich or deity who has a dark curse that turns wizards into nothics. The mechanics of how this happens, or what the curse might be, are left open to the DM to elaborate on if it is important to the campaign.
Best Answer
In fifth edition, the description of liches does indeed suggest that they are specifically wizards. But... well... fifth edition has said a lot of really dumb things, in my opinion, on the subject of liches. Requiring a major artifact in The Book of Vile Darkness, forcing archdevils or demon princes to be involved in each individual lich is ridiculous, and the whole “must consume soul energy” thing kind of... defeats the entire point of the lich. So I would not recommend taking fifth edition especially seriously here. It’s kind of full of bad writing—in my mind, a lich is defined by its selfishness and its obsession with collecting all the magic possible. A lich often seeks undeath precisely to avoid having to deal with others, fiends included. (No small number are no doubt an attempt to avoid paying up in some previous infernal bargain.)
Which is more or less how liches have been since the origins of D&D. While wizards are somewhat traditional—after all, wizards tend to be the ones obsessed with hoarding knowledge, which is precisely what a lich is about—there have always been liches from all kinds of spellcasting traditions, even ones you might think are incompatible like druids.
But they have always been spellcasters, and not just any spellcaster but fairly powerful ones. Again, collecting and mastering “all the magic” is kind of a lich’s basic raison d’être. The ultimate form of a lich was known as a “demilich” because they’re “only half there,” as they transcend this physical reality in search of more magic from beyond the limits of our existence.
In short, yeah, liches kinda have to be spellcasters. Non-spellcasters tend to favor other forms of undeath—becoming vampires, mummies, death knights, etc. In fourth edition, one could become a lich with “just” Ritual Caster, and no other magic, but even “just” Ritual Caster is pretty significant, especially considering that it also requires a fairly high level (in most editions of D&D, the minimum level to become a lich is around the halfway mark to the system’s highest level).