I think were saying essentially the same thing, but we're each just reversing the order of the terms used for the two back-up power systems?
Here's the way I see it break down based on the dialogue, 1st you have main/warp power, then 2nd you have the secondary/emergency impulse power, then you have the 3rd auxiliary/battery impulse power.
The "secondary systems' refers to impulse power with a functioning main energizer, the "emergency" impulse power is without the main energizer. In the first instance limited fuel has never been known to be a problem, but in the second instance, running low on fuel becomes a concern.
So then the auxiliary impulse power would be impulse power supplemented with "auxiliaries" or batteries, in which case limited fuel also becomes a concern.
In "Tomorrow is Yesterday", the bridge controls and main lights appear to be all dead but they're flipping switches and such. Spock reports that "secondary systems and impulse power" are the only thing functioning at that point with "auxiliaries" up to Scotty to turn back on. When the bridge lights and controls come back on we assume the "auxiliaries" are now on. Are "auxiliaries" akin to "auxiliary energizers"?
KIRK: Mister Spock.
SPOCK: Except for secondary systems, everything is out, sir. We're on impulse power only.
KIRK: Auxiliaries?
SPOCK: If Mister Scott is still with us, auxiliaries should be on momentarily.
Another thing to keep in mind, there may be a subtle distinction between impulse power and impulse engines/drive?
depending on the context Power could be either power to the impulse engine system via other sources, or it could mean power derived from the impulse engine system to power other systems, whereas impulse engines, or drive, refers specifically to the propulsion system. Disentangling these subtle distinctions may help avoid a lot of needless confusion.
Yeah, I too think there is a distinction between "impulse power" and "impulse engines/drive".
IMHO, Impulse power, like Warp power, is power that is stored for use by the ship's systems. Warp power energy is stored in the crystals while Impulse power is stored in I'm guessing a separate, impulse power storage system. Battery power is in a third and also separate storage system.
What if impulse engines work by taking allocated impulse power to drive an impulse field and converts/transforms a portion of it into matter fuel on the fly to be "fusioned?" and provide propulsive force? This would take in account the lack of large fuel tanks and also the ability to transfer additional energy into and out of impulse power storage. Also, I think energy converted to matter fuel could have remaining unused fuel reclaimed and converted back into energy for the ship...
Here's a brief list of impulse power being used in TOS:
"Tomorrow is Yesterday"
SPOCK: We are too low in the atmosphere to retain this orbit, Captain. Engineering reports we have sufficient impulse power to achieve escape velocity.
"WNMHGB"
Captain's log, Star date 1312.9. Ship's condition, heading back on impulse power only.
"Arena"
SPOCK: How about bypassing the transformer banks? Feed the impulse engines directly.
"The Squire of Gothos"
Captain's Log: ...However, by diverting impulse power to our sensors, we have made them operable, and we have detected one small area on the surface which seems relatively stable.
"Corbomite Maneuver"
KIRK: Now, Mister Sulu. Impulse power too.
"The Doomsday Machine"
SPOCK: At our present rate of consumption, we'll exhaust our impulse power long before then.
"The Apple"
KIRK: Then use your imagination. Tie every ounce of power the ship has into the impulse engines.
...
SCOTT: Sir, I'm going to switch over everything but the life-support systems and boost the impulse power, but that's just about as dangerous.
...
SCOTT: We're ready here, sir. All available power has been channeled into the impulse engines.
"The Immunity Syndrome"
SCOTT: Captain, we've only twenty six percent power reserves after entry.
KIRK: Do we have impulse power, Scotty?
SCOTT: I saved all I could, sir, but I don't know whether we have enough to get back out again.
"Elaan of Troyius"
KIRK: They're trying to force a fight. Scotty, what's our energy status?
SCOTT: Ninety three percent of impulse power, sir.
...
SULU: Captain, number four shield just collapsed. Impulse power down to thirty one percent.