Well, Green, you mostly got it already. AnotherB - i don’t think that Nix is around anymore. They showed the House of Heroes with only Harbinger when the Multiverse’s JL assembled there in recent Justice League.
The Monitor/s do change slightly in origin between multiverses; but they are aware of those changes and remain with the same powers and function.
1. The Over-Monitor and Anti-Monitor were created retroactively as having always existed when Krona looked through his viewer at the beginning of time and splintered the universe into infinite good and a single evil anti-matter universe. They were both killed during Crisis on Infinite Earths and remained that way until Infinite Crisis.
2. Infinite Crisis saw the recreation of the Multiverse by, yes, Alexander Luther Jr (with some help from Mr Mind) and it was through this “reboot” that the 52 Monitors and the Anti-Monitor were reborn. But again - they were recreated as a function of the multiverse; not in themselves.
3. Pre-Flashpoint
• The Monitors were destroyed through the actions of Mandrakk in Countdown and Final Crisis; reduced to a single Monitor Nix Outan. Mandrakk himself was an aspect of the Monitors since the Monitors were revealed to be a form of vampire; drawing energy from the ‘bleed’ between universes. The Anti-Monitor was killed by Superboy-Prime at the end of the Sinestro Corps War; and resurrected in Blackest Night.
Post-Flashpoint
• Nix Outan remained as the sole Monitor; despite the change from New Earth multiverse to Prime Earth multiverse. The Anti-Monitor, on the other hand, was reborn after Flashpoint as a being named Möbius; who was corrupted by the Anti-Life Equation. He again was destroyed in JL:Darkside War. These guys can die. They just don’t stay dead.
DC: Rebirth
•Both the original Over-Monitor and the Anti-Monitor Möbius reform; along with a third cosmic brother called the World Forger. The last ‘52’ Monitor, Nix Outan, vanished after Multiversity and was assumed to have become a part of the Over-Monitor.
The Anti-Monitor has, more recently, been re-exposed to the Anti-Life Equation and threatens the multiverse with destruction. Again.
Source material for the Monitors is found in:
PRE-CRISIS History
* Green Lantern Vol 2 40
* Flash Vol 1 #123
* Crisis on Infinite Earths #7 October 1985. Writer: Marv Wolfman.
NE History
* History of the DC Universe Vol 1 #1-2
* Green Lantern: Ganthet’s Tale
* Green Lantern Vol 3 32-35 (1992)
* Infinte Crisis #2, 5, 6.
* 52 #2-12
* Countdown to Final Crisis #49-37
* Final Crisis: Superman Beyond 3D #1
* DC Universe Vol 1 #0
PE History
(New 52)
* New-52 Justice League #40, 44.
* Green Lantern/ New Gods: Godhead #1
* Multiversity Guidebook #1
* Convergence #8
DC: Rebirth
* Dark Nights:Metal #4
* Justice League (2018) #16, 20, 22, 29
