The rules in the Player’s Handbook are clear - these types of familiar are a Warlock only feature. The MM is for DM fiat alone, and doesn’t indicate that the PH book is wrong in anycase. People on this thread, as stated, are just trying to have their cake and eat it.
I have to go with the cake-eaters on this one.
In multiple places the variant rules specify 'spellcasters' not 'warlock.' The fact that this is consistent with earlier editions has an impact too (at least to me). If they wanted to limit it to the warlock only, they would have done so.
Note that on pg 347 of the MM there's another variant rule that specifies 'ANY spellcaster' that can cast the find familiar spell...can be one of the creatures described in the spell...or some other Tiny monster, such as a crawling claw, imp, pseudodragon, or quasit.
I find 'any spellcaster' to be of perfect clarity.
Plus there is nothing in the warlock description that excludes other spellcasters from gaining these familiars. And any character can take the Ritual Caster feat to gain the ability to cast find familiar.
I also find 'variant' to be of perfect clarity. You can call it what you want, but variant rules are decided by the group in my campaign, not just the DM. Having said that, if you were playing in my campaign and I knew that it was important to you, then I would vote (and encourage others to do so as well) so that only warlocks could gain these types of familiars.
I would also point out that the rules as written in the PHB says your familiar, as defined by the find familiar spell, allows your familiar to take a special form. It's not actually an imp, quasit, pseudodragon or sprite, but just a fey spirit taking that form. Of course, the rules are currently silent on exactly what that means...
Having said all this, the real advantage is that a warlock can have their familiar attack for them, whereas all other familiars (conjured using the spell, including special forms) can't.
A special familiar that is serving of its own free will as described in the variant rules in the MM under imp, quasit, and pseudodragon can attack as much as it chooses, and would not expend the spellcaster's action to do so.
Ilbranteloth