Does a generic rulebook exist for the system behind Apocalypse World (Powered by the Apocalypse)?
There is no generic edition of Apocalypse World — that is, an edition with the setting stripped out and “just” the rules.
This is for the simple reason that the rules effectively are the setting, so there's no way to have “just” the rules. All the other Powered by the Apocalypse games were created by playing and studying Apocalypse World (and other PbtA games) to understand the design patterns involved, and then adjusting the rules to create rules that embodied a new setting.
However, one of the most well-respected designers of PbtA games — Avery Alder of Monsterhearts fame — has created a free PDF meta-game called Simple World (alternative link due to today's server issues) that is kind of like a playbook for a PbtA game. You follow the instructions in Simple World, and the result is a GM's Agenda, Principles, and Moves that are customised for the setting and themes you want to play with. It meanwhile instructs the players in how to create characters that suit this custom set of rules. This is as close to a generic set of PbtA rules that exist. Creating a game is fast (you do it together as a group while the players make their characters), so you can create a custom PbtA game and start playing in one session.
And once you've created a PbtA game with Simple World, you can always continue to adjust, tweak, and refine it, along with formalising some iconic character playbooks, ending up with a true stand-alone PbtA game of your very own design.
The closest I've seen to generic rules from a full game of Powered by the Apocalypse is The Bureau, which is a free game.