The Find Familiar spell describes a familiar
The Find Familiar spell description explains what a “familiar” is. It describes its properties — acts independently, can’t attack, can deliver spells, can be dismissed into a pocket dimension, etc.
The PHB uses “familiar” as a term, expanding its properties in some cases, e.g. Pact of the Chain:
Pact of the Chain
You learn the Find Familiar spell and can cast it as a ritual. The spell doesn’t count against your number of spells known. When you cast the spell, you can choose one of the normal forms for your familiar or one of the following special forms: imp, pseudodragon, quasit, or sprite.
Additionally, when you take the Attack action, you can forgo one of your own attacks to allow your familiar to make one attack of its own with its reaction.
So the Warlock’s familiar can be an imp, pseudodragon, quasit, or sprite, and it can attack.
“Variant: Familiar” describes additional familiar properties
Notice that imp, pseudodragon and quasit all have the “Variant: Familiar” block and there are no other creatures in the MM with the “Variant: Familiar” block. That block describes additional familiar features: first one is the 1-mile telepathic link, and second one depends on the creature type (for instance, Imp gives its master the Magic Resistance trait). That block does not describe any other familiar’s properties. It doesn’t explain anything that is already described in the Find Familiar spell, and it still uses the “familiar” term. It is logical to assume that it is the same “familiar” as the PHB describes.
From my understanding, “Variant: Familiar” expands base familiar description, like the Pact of the Chain does:
General familiar properties are given in the Find Familiar spell description.
If the spellcaster is a Warlock and they chooses the Pact of Chain feature, the familiar properties are expanded, giving more potent familiar as a result.
The familiar properties might be expanded even more, if the familiar is Imp/Quasit/Pseudodragon and the DM uses the “Variant: Familiar” optional rule.
It goes well with the “specific beats generic” rule:
Familiar can’t attack (generic) but it can with the Pact of Chain (specific).
The spellcaster can share familiar’s senses within 100 feet (generic), or 1 mile with several kinds (specific), providing the DM uses the variant rule.
So why in fact does it not?
Several answers (like this one) assume that in fact “Variant: Familiar” rule describes a sort of “another” familiar with completely different properties. It can attack, but can NOT deliver spells, and if it dies — it’s gone forever. How to obtain this familiar remains a mystery. It is still called a “familiar” though.
That seems purely homebrew for me, but is treated as official rulings. It is also being said these creatures cannot be found with the Find Familiar spell. What is the source of this assumption? What are the properties of this “another familiar”, and where they are described?
@SevenSidedDie mentioned that all “Variant: X” in the MM mean a somewhat special creature:
… a variant is just an alternative stat block for a given creature, so that it works slightly differently from the common variety of that creature.
The idea of “variants” is that, just like not every human is identical
A familiar variant of a creature is just the stat block to use for a creature that has somehow agreed to serve as a familiar for a spellcaster.
That actually supports the idea that Warlock should have the magic resistance trait from the imp, not contradicts it.
The Warlock summoned an imp as a familiar — therefore, that particular imp agreed to serve as a familiar. The DM uses the variant rule. Why shouldn’t the imp give the magic resistance trait, as the “Variant: Imp Familiar” describes?
It is just a spirit that takes a particular form
That answer explains it that way:
the Warlock’s familiar isn’t any sort of pseudodragon, let alone one with the variant. It’s a spirit that takes the form of a pseudodragon
But that reasoning does not work with the imp. Find Familiar implies that the imp familiar is an imp:
the familiar has the Statistics of the chosen form, though it is a celestial, fey, or fiend
So a fiend can be a familiar, and Imp is a fiend.