Types of Familiars
As the answer you quoted from and linked explains.
There are three kinds of familiar:
- Find Familiar - Does exactly what the spell says.
- Pact of Chain - Does exactly what the Find Familiar Spell says, with the exceptions that it can be an Imp, Pseudodragon, Sprite, or Quasit, and can attack if the warlock gives up his turn. This really is just the first type with some changes, and those changes might change again depending on the warlock's invocations.
- Variant Familiar - Is not RAW, but is an optional rule a DM may choose to include. The variant follows the rules of the Variant familiar sidebars for the given option.
A familiar will be one of these three types (or some homebrew), but only one of these three.
The variant rule is a variant that doesn't apply the conditions of Find Familiar, because Find Familiar is never cast. Here's the variant for IMP as an example (MM 69)
VARIANT: IMP FAMILIAR
Imps can be found in the service to mortal spellcasters, acting
as advisors, spies, and familiars. An imp urges its master to
acts of evil, knowing the mortal's soul is a prize the imp might
ultimately claim. Imps display an unusual loyalty to their
masters, and an imp can be quite dangerous if its master is
threatened. Some such imps have the following trait.
Familiar. The imp can enter into a contract to serve
another creature as a familiar, forming a telepathic bond with
its willing master. While the two are bonded, the master can
sense what the imp senses as long as they are within 1 mile
of each other. While the imp is within 10 feet of its master,
the master shares the imp's Magic Resistance trait. If its
master violates the terms of the contract, the imp can end its
service as a familiar, ending the telepathic bond.
That a player can't have more than one familiar at a time is a restriction of casting the spell find familiar. Variants do not follow those rules so that restriction isn't explicitly stated, so if your DM allows it, you may have multiple Variant Familiars. A few things here to remember:
The variant familiar is a Variant, and not stock rules. A DM may or may not choose to not allow your party/character to find any of such creatures (imps, quasits, or psudeodragons) that are willing to travel with the party in this way. Adventure League, for instance, doesn't include this variant rule.
The variant is may or may not controlled by the player, but could be controlled by the DM. So, DM has a whole lot more ability to reign in attempts at rules abuse. It also means the variant familiar can be a traitor.
That said, allowing a player the variant multiple familiars all with the ability to take its own full turn including attacking is a big deal. For instance, the imp is invisible until it attacks (so attacking with advantage), it can deal 1d4+3 damage plus up to poison 3d6 damage (or half as much). That is a max of 7+18 = 25 (* 2 on crit = 50 points of damage), or an average of 10 damage. A warlock can do this once per turn using his/her "improved" familiars, but it costs their action; to allow multiple free attacks per each with advantage and doing 10 damage each time is unbalanced and can be game breaking.
Since the contract in the variant is voluntary, it is also unlikely that an Imp, Quasit or Pseudodragon would be interested in serving a master that already is in a similar arrangement with another creature. Especially if the types of the would-be familiars differ. For example, Imps are devils and Quasits are demons, the two are bound in a eternal fight with each other, and would never voluntarily work together.
Jeremy Crawford apparently also ruled that the variant was only meant to be for NPC spellcasters:
The familiar variants in the Monster Manual are for monster and NPC spellcasters. PC spellcasters use the PH. -- https://twitter.com/JeremyECrawford/status/725190105451888640?ref_src=twsrc%5Etfw