Reading through, some flirt with my take on it, but none quite get there, so...
First, a reminder that Trek's history isn't ours. Divergence point is no later than some time in the 1800s, possibly earlier. This might seem obvious to some, but it bears repeating. They got the latter 20th century we were promised before Kennedy's assasination, before Nixon.
Just as one of the conceits of the Fallout universe is that the transistor wasn't discovered until almost a century after it was for us, the Trek universe had non-Newtonian propulsion and artificial gravity by the 1990s. Evidence: The Botany Bay. Matt Jeffries designed it as an "antique tramp space freighter". Episode dialogue establishes that craft like it were used as cryogenic sublight sleeper ships. One wonders if, say, a couple of those settled Alpha Centauri around the same time. I never understood why some of the ancillary materials hold that Khan and his people stole it and fled Earth. It always seemed to me an involuntary exile, hence the ship's name. Any rate, that ship has no visible thrust nozzles, rocket bells, maneuvering thrusters... No openings of any kind that I can see, actually. I've seen some postulate the rear pod is just a reactor. But I don't care how well you aim, you're still going to need to stop when you reach your destination. Or get yourself back on course, if the meteoroids don't also damage the wake-up interrupts.
So something like impulse existed in Trek in the atomic age. And, from the evidence of that ship, it's some sort of continuum-distortion drive. I liken it to a gyroscope -- like the kind one finds in science museums. Stand on a well-lubricated turntable, give the bicycle wheel a good spin, pick it up. Hold it straight, you don't move. Tilt it this way, you start to turn in the opposite direction, and vice versa. Some sort of superconducting possibly-unobtanium whatsit spinning in a plasma field generated by the atomic pile (and, later, fusion reactors), and when it gets deflected one way or the other, it creates a gravity "incline" the ship rides.
I loved Diane Carey's ability to create interesting characters, write good dialogue, craft compelling narratives... It largely made up for the fact that she sucked at a lot of the technical details. One thing she did in Final Frontier (the book, not the movie) that I appreciated at the time, then denigrated as double-talky nonsense, then more recently realized it worked better than anything else, was her explanation -- from one character to another -- how impulse worked, and it was, essentially, that. Also that it was an holdover acronym that had become a work in its own right, like laser. I.M.Pulse. Internally-Metered Pulse drive. Regulated power surges determine the degree of continuum distortion. The latter character opines that it sounds like he's describing warp drive, to which the other engineer scoffs that warp is as far above impulse as impulse is above walking. Orders of magnitude.
Which got me thinking about Cochrane's breakthrough. If he was born on Alpha Centauri to Terran emigreés, Kirk's description of him in "Metamorphosis" holds. If high-sublight travel was a thing already as he was in his adolescence... What if, I can see him thinking, instead of varying the firing of this impulse coil, I take a bunch of them in a stack and fire them in sequence or other combinations? Early tests showed off-axis instability, so he shrank them and added a second stack to balance the first. The initial flight-test vehicle didn't quite manage it, but he was pretty sure he knew what went wrong. When he got to Earth he would just tweak some things and try again. Then he got a Third World War in the face and had to come up with a Plan B.
Anyway. As with Cochrane's warp ship, and earlier sublight ships, I figure Human FTL ships were powered by fusion reactors for a goodly while before antimatter was easy enough to make and safe enough to store to make it a practical power source for starships. And, even as antimatter came into use, the older drive style still consumed far less power for a sublight propulsion system than the big warp engines, so they stayed in use, typically, by the time we come into the story, with their own backup fusion reactors to power them in case of emergency or separation or while the antimatter reactors are shut down for repair or maintenance. An impulse fusion reactor could still probably power a small ship up to low warp, as in the early days, but for anything meaningful you need a power source with more oomph.
So when Scotty notes the Romulan Bird-of-Prey is powered by "simple impulse", it doesn't necessarily mean sublight-only -- just that the Enterprise will definitely be able to outrun it at warp... if it's visible.
It's one of the things I love about Andy's design of the movie Enterprise impulse deck. The "deflection crystal" is using warp plasma to power the impulse engines, too, even though they also still have backup fusion generators.
And the location of the impulse deck, the black rectangles on the original, the slatted panels on the later ones... Primarily heat-exchangers, protected but exposed to space. But also, yes, exhaust vents for reaction products, to prevent overpressure. Not much -- certainly not enough to push the ship -- but enough it makes sense to point them rearward along the vessel's typical direction of flight. The fusion plants alone don't warm them up enough, but when they're running under warp power, and/or at high sublight, they'll definitely be glowing orange-red.
How's that?
I've been mindful of all the stuff in both this series and TNG saying over and over that our (23rd/24th century) records of the time are fragmented and incomplete, due to everything that happened in the Trek universe between WWII and WWIII. I wonder what Kirk and Spock's respective commentary would have been on the LEM being actually used to land on the moon. Remember also, Kirk's interest in history was more 19th century -- pre-spaceflight. He was even the better part of a century off with IDing the Botany Bay (and subsequent remarks to Khan) as a DY-500 versus a DY-100. His comment is correct and doesn't negate using hibernation for interstellar -- just says that hibernation was needed until ships got faster over the next decade.
?). During the escape maneuver, the Cube was able to close in on the Enterprise when she was at Warp 3. Did the Cube really travel at the speed of light using a warp drive to run at warp one? Later, Balok's ship towed the Enterprise at 0.64 c. (It would take 7-8 months just to travel one light year, so, their designation must have been really close.) Balok may have been using a warp drive system to tow at sublight speeds since their Cubic demonstrated the First Federation had FTL capability. When Kirk broke free, the Enterprise was using only warp drive initially (a warp drive tug of war?), then threw in impulse drive at the end to provide enough power to "fake-break" Balok's drive system.
