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.