Hmm... I think that tells us the Stephen E. Whitfield was told that the impulse engines are fueled by entirely other means than the warp drive and that battery power is again some entirely different system.
The use of the phrase "controlled fusion of matter and antimatter" in this case is not nuclear fusion in the usual sense, but, rather, simply the introduction of matter to antimatter in a controlled manner which allows the resultant energies to be collected and harnessed. This is quite a different thing from sticking two hydrogens together and making a helium and a spark of extra juice.
OTOH, the Impulse engines has some sort of fuel that it burns to do it's thing, and, provided that fuel is exhausted, there is a different system called "battery power" which can be useful for a week or so, depending on the usage requirements.
As I see it, your misunderstanding is that all three are interrelated and use the same basic energizing material.
Where my mind is at on this (for the moment anyhow) is that the warp drive is powered by matter-antimatter annihilation reactors which may or may not utilize fusion-related technology to do it's thing. The impulse engines are powered by their own heavy duty fusion reactor which operates totally independently from the M/AM reactors. (BTW, I'm still in the camp that impulse drive is a smaller scale subspace driver, similar in concept to the warp drive... not a giant rocket thruster... but that may be beyond the scope of this thread... I don't want to derail this.) Then there is a line of (i.e. a "battery" of) small fusion reactors which provide emergency power.
Now, going by TMoST, we might wonder how often Enterprise
has to stop for gas. If the ship can run on impulse continuously for only a month, then are our heroes going to a service station every month or two? Or are we to assume that the warp drive is used almost all the time and sublight maneuvers are relatively rare and only use the impulse drive for a few minutes at a time, which might drag out the refueling time by quite a very long span. In fact, if you are only using it for an average of 15 minutes a day, you might go seven or eight years without needing a pit stop!
Regarding "Auxiliary Power," maybe it's the sum of all the ship's fusion reactors going at once? The Impulse reactors and the emergency batteries together could create a pool of energy which could be drawn from as needed when the main M/AM reactor in Engineering isn't up to spec for whatever reason...