All we can determine from dialogue is that the Enterprise had at least three reactors on board of indeterminate function, although the impulse engines are apparently powered by fusion reactors.
When there is talk about what powers the warp drive, aside from vague generalities about the nacelles, with matter and antimatter being involved somehow, the few instances of direct references (interestingly, in two episodes either written or directed by John Merydith Lucas, specifically, "Elaan of Troyius" and "That Which Survives"), we have the "Matter/Antimatter Reaction Chamber" and "matter/antimatter reactor".
Note the use of the singular in the reference.
It is specifically from those references, and those from other episodes that tend to support that model, that the folks who did the design work on later Trek shows derived their model of a single intermix chamber, or "warp core", providing the power to operate the warp drive, with other secondary reactors, presumably fusion, to provide supplementary power to the other systems onboard.
Also note that this model is put in place during the development of Phase II, presumably with the involvement of Matt Jefferies. In fact, a lot of his redesign of the Enterprise seemed to be of the nature of clarifying things that were a little vague in the original ship, like where Engineering was, which way the bridge faced, and how the power systems were set up.