Below is my take on the original TOS/TAS timeline before things got retconned and hopelessly complicated by later series/movies, apologies for the length, this was lifted directly from my notes.
Uses of “space warp” in TOS/TAS
Zephram Cochrane “discovered” the “space warp” and was believed to have died c.150 yrs BT (Before TOS) yet he is known as Zephram Cochrane “of Alpha Centauri” despite being referred to –and acting like- a human from Earth. So perhaps Cochrane “discovered” the space warp when he was still quite young, say around the year 2018 when Lt. Marla McGivers said sleeper ships like the Botany Bay became no longer necessary for use within the solar system.
So it was at this point that limited interstellar travel became feasible, albeit still requiring cryogenic sleep for months/years beyond the solar system?
This, would not only explain how the Valiant “from 200 years ago” (WNMHSGB) got out into deep space but also how the Terra-10 colony (TTI) was established so far from Earth. (Especially if they were both using early space warp drive combined with sleeper ship technology)
Below is relevant dialogue from “Metamorphosis”
KIRK: Zefram Cochrane of Alpha Centauri, the discoverer of the space warp?
COCHRANE: That's right, Captain.
MCCOY: But that's impossible. Zefram Cochrane died a hundred and fifty years ago.
SPOCK: The name of Zefram Cochrane is revered throughout the known galaxy. Planets were named after him. Great universities, cities.
KIRK: Isn't your story a little improbable, Mister Cochrane?
COCHRANE: No, it's true. I was eighty seven years old when I came here.
If Cochran was 87 yrs old when he was lost to the pages of history, and that was a 150 years before the Enterprise’s five year mission, then Cochrane would have been about 37 years old in 2018!
Kirk: “Our space warp ability gone” WNMHGB
In neither case is there an association with warp drive per se.
On the other hand…
Dialogue in TOS/TAS suggesting that some relatively recent and significant breakthrough in propulsion technology (c. 30 yrs BT) -relating to time- has occurred (And which the Enterprise/Constitution class is among the first so equipped)
SPOCK; “S.S. Columbia, disappeared in that region approx 18 yrs ago”
TYLER: (to crash survivors) “And you won’t believe how fast you can get back. Well, the time barriers been broken, our new ships can…” TC/M
So perhaps the newer tech is an improvement or addition to the older more limited space warp tech, one which overcomes some limitation with the latter which had come to be known in general parlance as “the time barrier”?
The above has interesting implications for seeming contradictions between TOS and TAS as to when “warp drive” technology was first implemented.
SARAH APRIL: “the first chief medical officer on a starship equipped with warp drive”. (It has been generally been assumed that the starship in question was the Enterprise, since her husband -Robert April- was the ships first captain.)
On this basis it seems that Sarah April uses the term “warp drive” as a reference to the more recent time warp tech. suggesting perhaps, that it is the time warp tech that has the drive or field propulsion aspect associated with it? So perhaps Cochrane’s earlier “discovery” of the space warp was for a static warp field with no drive aspect associated with it and therefore merely allowed for the impulse engines to “cheat” the light speed barrier?
If we accept that the SS Valiant was an early space warp/sleeper ship combination, then it would still use impulse (rocket) power for propulsion, this is consistent with Kirk’s statement that “the old impulse engines weren’t strong enough” to resist the magnetic storm pulling it off course.
Then there’s this from “The Time Trap”;
SCOTTY “There’s the old Bonaventure; she was the 1st ship to have warp drive installed”. (A ship that looks a lot, perhaps tellingly, like the Enterprise)
Interpreting this is somewhat problematical; on the one hand, the design of the Bonaventure suggests that it could be a prototype vessel for the more recent “time” warp innovation that resulted in the production-line ships like the Enterprise. This is supported not only by its overall design configuration but also by the “NCC” on the nacelles, as well as the fact that a female member of the Elysian council is wearing what appears to be a “Cage era” Starfleet uniform and therefore may be the captain or one of the crew. On the other hand, Spock’s statement about “The crew's descendants may still be living, Captain.” suggests that it is perhaps the older “space” warp tech that is being referred to after all?
In my opinion, the preponderance of evidence here seems to point toward the conclusion that the Bonaventure should be considered as equipped with the newer time warp innovation and Spock’s statement should be taken in light of a similar statement he made in WNMHGB regarding one of his “ancestors” who married a human female.
In any event, note also that the use of the terms “installed” and “equipped with” in each case might imply some contraption that can be added to existing starship’s internal engineering systems, and therefore not something requiring a wholesale redesign of existing starship hull configurations.
Other specific uses of “time warp” in TOS/TAS
PIKE/SPOCK: “Our time warp factor” TC/M
-Suggesting that the later oft used term “warp factor” is, in fact, an abbreviated form specifically referring to the time warp as opposed to space warp
The association with warp factors in TC/M might suggest that the warp scale formula (given in TMOST/TWG) -which has been criticized for not yielding the kinds of speeds to cover the distances seen in some TOS episodes- is therefore actually the old, slower speed of C cubed (space) warp factor scale and that the newer time warp tech augments or modifies this scale so that the newer (time) warp factors are considerably faster, but not, apparently, in any consistent integer value? Alternately, perhaps the warp factor formula is for the time warp drive after all, but that it must be cross calculated with the older space warp velocities to arrive at a true ETA at any given time? The new scale factor would initially be distinguished from the older by calling it a time warp factor, at least until the new scale became standard, after which it would just be called warp factors again. Be this as it may, it should be kept in mind that the warp factor scale was never explicated in onscreen dialogue; therefore we are free to ignore altogether it if we wish.
To sum up, the seeming contradiction between TOS and TAS can therefore be resolved if we understand the term warp drive to mean time warp tech, either by itself or when wedded to the earlier space warp tech.