The AI is a single entity. Calling the command stations "servants" is akin to calling the left and right hemispheres of your brain your servants. Destroying both Home Command Centers destroys the AI's brain.
Hm, yeah, okay for "one entity".
However, the Core is supposed to reside in NGC 224. The Exodian Blade campaign tells that rather explicitly:
The Core has escaped. I was too late, and its exodus via the Pyletzan Conduit was successful.
So the things it left in the Milky Way are... well, I call them servants, but I read the journals again and found the origin of my mistake: the Second Exile talk about it's "Brother's servants" and the "Servants he crafted", but it seems they are the numerous AI ships rather than the two AIs.
I can vanquish many of my brother's servants, but my own power will very likely not be enough. Send forth whatever you can muster to ensure I complete my mission.
So, no: the Core is far from vanquished in a vanilla ending. Only the two "things" it left in the Milky Way are destroyed and it lost its ability to easily access our galaxy (the exogalactic wormholes, the famous "Pyletzan Conduits" by which the Core and the Blade were banished).
Bring me to one of the Conduits, what you call "Exogalactic Wormholes", that is still guarded by one of his main control centers.
Ah... "Main Control Centers". But not "Main Center" as in "half brain", rather "Remote Controlers"... What I called servants. If destroying the two "AI/servants/stuff" in the Milky Way was enough, the Blade could have done that. But if it crosses a Conduit (an exo.wormhole), everything ends. I interpret that as "the Core behind the conduit in NGC 224 is far more important and central than the two Control Centers in the Milky Way".
So yes, I call them "servants", and I feel it's right. (It's not because you changed pseudo and avatar that I don't respect your "Arcen Lore Chair" anymore. I just read, deduce, quote and explain.)
What's your opinion on that?
And while you're here, tell me: who/what is Kyr-Sun? I personally believe it's a military/scientific organization from Earth, but there is especially few hints in the journals.
But if the AI is busy in some other galaxy and is so huge that it's operating in multiple galaxies, why is it entirely reliant on two command stations over here for its entire operation? That seems weird.
Well, there are Subcommanders, that are special OCStation with a "personality", in the game. I can see them as "threads". In a similar way, I can easily imagine the Core delegating the management of the Milky Way and the "Human affair" to two powerful "servants" he crafted with the Artificial Intelligence and Fleet Control technology given by Langstrom and Artain (and maybe that mysterious Kyr-Sun organization).