Okay, until we hear some solid background information on the alternate engine core painting, I'd say it's up to speculation and here are my 0.02 $:
- This painting is post-Minor and post-Phase II because it features no longer the "Minor spheres" but these cylindrical and connecting segments instead which we saw in he final film
- We are probably looking at the first engine core extension painting intended for a scene that didn't make it into the film. Either Bob Wise wanted to shoot Kirk's entry differently or the original plan (a camera pan from the lower core to the upper engine room level / physical set) didn't work out because it became too obvious that the extension was fake (this was still the "analog" age, so how this would have looked on film depended entirely on shooting it first and then evaluating the footage)
- Instead, they decided to shoot the engine core extension from a higher point of view with a moderate camera pan that wouldn't betray the intended illusion ("play it safer") and it worked. Possibly a late decision, thus Ben Resella didn't have time to include as many cylindrical connecting segments as he did in the first extension painting.