We can identify at least one of the Mr Scott's
Okudagrams in ST4, and verify that it doesn't have the long word "TRANSWARP" where SJ puts it.
More specifically, the side view of the ship here (the upper individual monitor to the left of Spock's head) is otherwise identical to the one where SJ "reproduces" the bottom line label as TRANSWARP SUBSYS 7 4516, but the bottom line label in the movie is actually shorter than that - that is, it terminates more to the left, in comparison with the random column of numbers above it.
This particular Okudagram no longer appears in the Trekcore HD screencaps of ST5 or ST6; similarly, the side view with warp field curves (labeled TRANSWARP GEOMETRY 4 1190 by SJ) is absent.
In the end, none of the HD images reveal the word "TRANSWARP", and indeed most of the Okudagrams in the latter two movies seem to be labeled only with random numbers, not with descriptive words.
This of course does not debunk the idea that the technology called "transwarp" became known as "warp" as soon as it was proved to be practical.
Nor does it debunk the possibility that the theory underlying the Excelsior
experiment was proven completely erroneous by the first attempt by the Excelsior
to actually explore its supposed transwarp envelope. Captain Styles seems particularly moved by the computer's declaration "all speeds available through transwarp drive", as if he had never heard the boast before. He also feels he will only be breaking the earlier speed records of the Enterprise
"tomorrow"; clearly, neither he nor anybody else has yet flown the Excelsior
or any preceding testbed at speeds exceeding those of Kirk's ship.
Why Starfleet would put so much faith in a faulty theory that they base the design of a huge starship on it, we never learn. But perhaps we misunderstood? Perhaps the Excelsior
was a previous investment, and the transwarp experiment was only brought aboard later on, at relatively low cost?