My pet theory is that it's the same thing as the pair of cylinders on the engineering floor in TOS: a location for inserting the all-important dilithium crystals into some sort of a carefully shielded dumbwaiter/waldo system that takes them to the heavily armored reaction chamber in the bowels of the ship.
Prior to TMP, we didn't get to see any non-shirtsleeves engineering environments. Doesn't mean the whole kit'n'kaboodle from TMP wouldn't have been in place in TOS already. It's just that in normal conditions, nobody would enter this facility and risk radiation exposure. During a test flight after a refit, the presence of radiation-suited personnel down there would be a must, though. During a training cruise, engineer trainees would be working there as well. But the "real" engineering work would still happen in shirtsleeves, up in a facility looking remarkably like the TOS engineering set.
So the room from ST2 is a dilithium delivery system access point added in the non-shirtsleeves environment (possibly with a shirtsleeves point retained in the upper decks as well). Because the facility is already in a high-risk area, it's configured differently from the TOS access point: it provides more direct access, but is in turn surrounded by that fallout-containing plexiglass which would be useless in the TOS layout but serves a purpose in an environment that already features protective suits and other such precautions.
By accessing the dilithium delivery system, Spock would be able to extract the crystals, revitalize them manually, then redeliver them to the reactor to get the drive working. Normally, a device called "energizer" would revitalize the crystals (just like an "energizer" was used in TOS "Alternative Factor" to nurse back to health a set of crystals harmed by the universe hiccuping), but in this case the "main energizer" was "out", like Scotty said. So Spock had to do it by hand, probably by finding a new, still working facet from the damaged crystals.
Perhaps the energizing in "Alternative Factor" also involved finding new, not yet "spent" facets of the crystals? It's just that the TOS crystals would be microscopically small and embedded on those ping-pong paddle devices in vast numbers; Starfleet could extremely seldom afford macroscopically large crystals. The energizers would realign the microcrystals - but Spock would be skilled in the art of realigning by hand the slightly larger crystals Starfleet of the 2270s could afford. And we know that by the time of TNG, macroscopic crystals are commonplace and can easily be grown or regrown...
Howzzat for a rationalization?