If I recall TWOK correctly Scotty says "just batteries. You'll have auxilary power in a few minutes". Since according to the red dots in the ship's schematic on the bridge, Reliant
hit mulitiple spots on the Enterprise
and might have severed several power conduits from the batteries, too, that required a manual fix before auxilary power (from the batteries) could be distributed to the systems in need of power. I'd like to think that "battery", "auxilary", "emergency" and "reserve" power are one and the same thing.
The one thing that causes me headaches is Scotty's "battery power" remark in the context of the impulse drive in "Mudd's Women". If you have nuclear fusion reactors you'll only need to ignite them once. Since we've seen exotic force beams rendering the antimatter inert, we can't exclude the possibility force beams that would render a nuclear fusion reactor inert, thus you have to use the (extremely reliable as you suggested) batteries to "restart" or "re-ignite" a fusion reactor which is probably what had to be done in "The Naked Time" and "Court-Martial", not to mention you have to have one outrageously reliable energy source to keep the magnetic containment fields holding the antimatter intact.
However, in "Mudd's Women" there was no such exotic event that either switched off the fusion reactors or forced them to do that (it would have been plain stupid). If "battery power" starts a nuclear fusion reactor or propulsion engine there is no need to keep draining the battery supplies once the "re-energizing" fusion process has been set in place. Frankly, I'd like to file Scotty's remark as yet another one in my secret list of "Scotsman Exaggerations" ("Look, I can make a fusion drive work with only battery power" Reminds me of the "cold fusion" hoax).
But then again, we also have to wonder if / why they didn't use the "secondary power systems" (i.e. fusion reactors) for transport down and up to Rigel 12. Could the lack of dilithium crystals have something to do with the secondary systems, too?
It's apparently difficult to make a clear separation between "batteries" and "fusion reactors", especially if you assume that the energy required for the transporter system will at least require fusion power.
I wholeheartedly agree with the basic power infrastructure: main power, secondary power, emergency power.
It's the details I'm not entirely sure, yet. "Energizer" by definition suggests a device that supplies power. Apparently it's neither the "batteries" nor the "matter-antimatter reactors" of the Enterprise
(i.e. why call it "batteries" when it is in fact the "energizers"?)
Again, if these terms would hint a different power output ("battery" = fusion power without crystal conversion / "energizer" = fusion power with
crystal conversion) we might be closer to a satisfactory answer but obviously I'm getting ahead of myself or part III of this treatise.
The way I understood events in "The Naked Time", however exotic and far-fetched, they needed warp power to counteract these gravity effects of contracting Psi 2000 and impulse power would have only / did buy them time.
That piece of dialogue from "Tomorrow Is Yesterday" you mentioned is very interesting. Apparently, impulse drive belongs to the "secondary systems" (if I recall correctly it is the "secondary drive") but then there are still these enigmatic "auxilaries".
To avoid a crash the ship's computer apparently took over and channeled all the energy output (including fusion energy sent through dilithium crystals, I think I hear the TMP Enterprise
knocking at the door) into the impulse engines. Only when the ship had managed "escape velocity" some energy was rerouted to the auxilary systems to power inertial dampers, life-support and artificial gravity that had been previously neglected and with good reason - the first priority was to avoid a planetfall crash at all