I appreciate any insight to this question, including suggestions for other terms to learn about first. I am self-taught with regards to set theory and not a mathematician, so my question may not be well-formed. Thank you for your patience!
To get to this question I have read numerous Wikipedia articles (set theory, ZFC, ordinal numbers, limit ordinals, etc.), spoken with some grad friends who study math, and read a paper and part of a book by Thomas Jech.
I wanted to know if there exist set theories in use that do not use an Axiom Schema of Separation. My understanding is that in response to Russell's Paradox (itself a critique of Frege) in ZF we have the Axiom Schema of Separation instead of the Axiom Schema of Comprehension (as Jech teaches in Set Theory).
I've been curious about alternative set theories and have found the Wikipedia pages for NBG (Neumann Bernays Godel) and MK (Morse-Kelley). Am I correct in thinking that both of these have something equivalent to the Axiom Schema of Separation?
I am specifically interested in the following notion: is there a set theory in use today that does not rely on the Axiom Schema of Separation to construct its existents*? It seems the power of the Axiom Schema of Separation is to get out of Russell's Paradox by assuming there exists some other set, say $Y$, that one then 'cuts' from in order to construct the set you really wanted to bring into being (say, set of all $x$ with some property).
I'm curious if there are set theories used today that don't do this move, and if so, how do they deal with Russell's Paradox? In short, I'd love a list of alternative strategies that set theories use to create new sets. (Edited after reading Asaf Karagila's kind answer.)
*Existent is a word borrowed from philosophy. It just means "thing that exists." As I understand ZFC, everything in it is a set, so if we were just talking about ZFC instead of "existent" I could have said "set." But, in Morse-Kelley it seems like sets and classes both exist, so I use the term "existent" instead.