It's almost correct, but not quite.

There are two separate, unrelated ways to get a familiar in D&D 5e, and you're trying to combine them.

The first method is by use of the Find Familiar spell, which allows you to summon a celestial, fey, or fiendish spirit that takes the form of any of a list of creatures. This list is expanded by the warlock pact of the chain. This familiar is perfectly obedient, can be resummoned when it dies, can be hid in a pocket dimension, deliver touch spells that you cast, and everything else specified in the spell description. This method gives you a familiar with the basic stats of the chosen creature, not the "variant: familiar" traits of the chosen creature (unless your DM chooses to have that creature type appear).

The second method is by finding a quasit, imp, or pseudodragon that has the "variant: familiar" trait (which is 100% up to the DM), and enlisting it as a familiar by interacting with it. This familiar has only the traits listed in the stat block for that creature, including the variant traits, but none of the traits of familiars given by the Find Familiar spell. No pocket dimension, no touch spells, no limitations on what actions it can perform, and if it dies, it's just dead. This type of familiar is an NPC controlled by the DM (much like a hireling or other follower), and is only as obedient to the PC as the DM says it is, using the MM entry as a guide.

Now that we've established how things actually work, we can address your real concern. Your warlock can't yet communicate at a great distance, but he can soon. There is a warlock invocation available to him called Voice of the Chain Master that does the same thing, but with unlimited range on the same plane. As you've realized, this ability has some incredible potential, especially for scouting.

It's not any more powerful than other options and should not be limited.

Let's compare it to some other invocations. There's one that lets a warlock cast Disguise Self as at will. This would let him see a guard, and appear exactly like that guard and just walk around the enemy camp unimpeded. Or maybe impersonate the leader of the camp and just take it over without even a struggle. There's another invocation that lets the warlock cast arcane eye at will, which gives you a way better scout than an easily killed creature. A familiar, even an invisible one, still has to succeed on a Dexterity (stealth) check to avoid being heard and then easily killed. An arcane eye does not.

So, in order to fully utilize this scouting ability, your warlock has to pick the chain pact and spend one of his few, precious invocations, both of which are huge opportunity costs. He deserves something in return. This something is you not limiting it. It already has a flaw in still being able to be killed by anything that hears it, or smells it. That's right, just about any pet wolf is going to be enough to catch this familiar. It doesn't need any more limitations.

No, a Warlock with the Pact of the Chain feature does not receive Magic Resistance if they choose a Quasit, Imp, or Pseudodragon. In the first place, a variant rule is only in play if the DM chooses. However, even if the DM decides that pseudodragon familiars (for example) are a thing, it still doesn't benefit the Warlock.

Some pseudodragons are willing to serve spellcasters as a familiar. Such pseudodragons have the following trait.

So "some pseudodragons" will be familiars who share Magic Resistance with their masters. The Warlock, however, gets their familiar from the find familiar spell, which says:

You gain the service of a familiar, a spirit that takes an animal form that you choose: [...]

The Pact of the Chain says that:

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.

In other words, 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 (and can take a different form anytime the Warlock chooses). The spell does say that:

[...] the familiar gains the statistics of the chosen form [...]

However, the statistics of a creature are defined on pages 6-11 of the Monster Manual. The short definition is that, if you look at a creature's entry, it's everything in the yellow box. This doesn't include variant traits like the Familiar trait - those are in green boxes off to the side.

All of that aside, a generous DM could, of course, allow a Pact of the Chain familiar to provide its master with Magic Resistance. It must be pointed out, though, that this is making the Pact of the Chain vastly more powerful than it normally is, so DMs thinking about it should consider carefully.

An even more generous DM could allow any player to gain the service of a creature with the familiar variant. This, too, should be considered carefully - it's effectively giving the player a more powerful version of the Pact of the Chain feature for free.