That is an interesting read as well.
IMHO, I do think there are some differences or tiers in power generation systems.
The main power generation system
1. The Space Warp Engines via M/AM reactors in the nacelles
2. The M/AM reactor in the engineering hull
The secondary power generation system
1. The impulse engines (fusion reactor? - "The Doomsday Machine")
2. Auxiliary power (fusion reactors?)
The emergency power generation system
1. The batteries
All those sources then plug into the "energizers" to be distributed to any system on the ship that can be energized aka powered. I think that:
1. The crystals in TOS operate as a super buffer or capacitor that also has the side-effect of creating more matter-antimatter fuel.
2. Without the crystals, the energizers would then have the power source energies just go through the bypass circuits. In other words, if the bypass circuits were not burnt out, the Enterprise could still fly at warp speeds and operate her equipment but at a much reduced power level (no buffering) and likely with the possibility of running out of matter-antimatter fuel (no regenerating.)
On "Mudd's Women", this is just a more subtle interpretation with the missing line before Scotty's "it'll be a shaky orbit".
From the dialogue, I read it as:
1. We've been traveling on the auxiliary impulse engines for a while with the batteries supplementing power (which also explains why they're running low at the end since they've been using them for a while already).
2. Prior to "pre-orbit" they ran out of impulse fuel or power and were running exclusively on batteries for the remainder of the episode until they get crystals.
3. The batteries are powering "something" to put her in orbit and hold her there. It is likely the impulse engines or could be some other propulsion system.
KIRK: Captain's log-- Stardate 1330.1. Position, fourteen hours out of Rigel 12. We're on auxiliary impulse engines. Fuel low, barely sufficient to achieve orbit over the planet.
FARRELL: Preorbital course locked in, sir.
KIRK: That's the last time I'm giving an order twice, gentlemen. We're down to battery power, and we're low on that.
SCOTT: It'll get us to Rigel 12, sir, but it'll be a shaky orbit.
Could then the phrase "auxiliary impulse" in "Mudd's Women" mean power to the impulse engines provided by auxiliary power and "emergency impulse" in "The Doomsday Machine" mean power to the impulse engines provided by emergency (or battery) power?
We also have this bit from "Tomorrow is Yesterday" where Scotty is using impulse power ("a secondary system") to fly the ship in the atmosphere but were distinctly separate from auxiliaries (or auxiliary power systems?).
SPOCK: Except for secondary systems, everything is out, sir. We're on impulse power only.
SPOCK: If Mister Scott is still with us, auxiliaries should be on momentarily.
Also, a slightly puzzling piece, in "The Naked Time", there must be an untold reason why the impulse engines were ignored in favor of the warp engines to be turned on. From the dialogue, either warp or impulse engines could've saved them so does that mean:
1. Impulse engines take even longer to bring up to power from a cold start than warp engines given that auxiliary power or possibly batteries could've powered the impulse engines?