I know that Find Familiar places limits on what actions a familiar can take in combat and when it attacks and that they are different from those of the Best Matter Ranger's companion. The Warlock Pact of the Chain is also different to my knowledge. 

But how does it work for a wild pseudodragon that you find or if a character/enemy has a Gazer familiar? So the same rules apply even though they aren't magical summons?

This is something you would want to work with your individual DM on.

There is a variant rule in the Monster Manual entry for the Pseudogdragon that simply says "Some pseudodragons are willing to serve spellcasters as a familiar." and grants them a Familiar trait in their stat block that details the benefits.

That variant rule does not require the magical summoning, it's a willing bond between the character and the pseudodragon. This is why the individual DM needs to check in. How you convince the pseudodragon to bond with the character is something the DM would control in their game.

The Find Familiar spell is irrelevant to any familiars you get through other means. Just ignore it. Whatever rules are granting you the familiar (e.g. the Variant: Familiar sidebar in some monster stat blocks) will tell you if it has any special powers.