I apologize for my overly long response to EVERYONE's comments. I kept putting it off and the discussion kept getting longer. haha.
LOL, yeah it's an interesting subject with lots of detail. Good discussion all around

I agree that they are probably the same system. However, it seems that the function of the lithium crystals is different from the dilithium crystals. In Mudd's Women the lithium crystals are destroyed because excessive power is being pulled through them. Whereas dilithium crystals, for the most part, seems to be described as the source of power. Now perhaps, in-universe, these lithium circuits got switched out for something else (more reliable) later on. But at least for me, they seem distinct enough in function that I'm going to consider the dilithium crystals and lithium crystals to have two separate functions, and they probably existed alongside each other in the "converter assembly."
IMHO, the Lithium Crystal and Dilithium Crystal circuits work the same way. We have the dialogue about pulling power through the crystal circuits but there is this one line after 3 crystals are blown:
From "Mudd's Women":
FARRELL: We're clearing the asteroid belt now, Captain.
KIRK: Deflector screen down, Mister Farrell. Save the power.
That line only makes sense if the last crystal circuit is also providing power rather than just transporting power, IMHO.
Do we even know what the energizer is? I can only find two references to it that aren't from the Movie Era. The first is in relation to the circuitry that re-amplifies the dilithium crystals. The second is when the "main energizers" are damaged by the Doomsday Machine. Are they even the same thing? There seems to be little information on what a system known as an "energizer" does.
In real-world terms, an "energizer" supplies energy (electrical, etc). In "The Alternative Factor", the crystals were in an energizer presumably either providing energy to the ship or to the crystals or both. In "Doomsday Machine", the main energizer going down also meant the loss of warp power. Since the warp engines depend on the crystals then we can connect the main energizer as containing crystal circuits that are connected to the warp engine usage.

Mmm. Not quite convinced of that. But I already touched on that above. Though I don't think we can reliably use TWOK here because the systems were probably different. Perhaps they realized that the DCCA setup wasn't the best, and increased the battery capacity to allow firing of phasers from batteries. Plus the conversion from caseless to vcased torpedoes allowed them to be fired with very little energy expenditure.
In my head canon I imagine that Kirk's Enterprise underwent a few upgrades and refits over the years. Transitioning from Lithium to Dilithium energizers, installation of photon torpedoes, upgraded shields post-"Balance of Terror", engine rooms redesigned over 3 seasons, new bridge module post-WNMHGB, etc...
Lithium crystals were only in one episode.
You are correct. I had lumped in the "Lithium Cracking Station" from WNMHGB where they adapted the station's power packs to the Enterprise's engines.