And where does the notion that the shuttle flying ahead of it was towing it come from? I always just thought they were just leading them.
That's an interesting issue you are touching here.
- In TMP it seemed like Scotty was manually steering the pod or at least making occasional course adjustments
- In TWOK it looked like Sulu and/or Uhura were making occasional adjustments
- However, in TVH it really looks like they are all just standing without anybody interacting with the pod's controls.
Update: In TVH it was the first time they attached an "Operating Instructions" label onto the "door".
Lee Cole made several of such labels for TMP and those were recreated (i.e. not accurately reproduced) in the TMP Peel-Off-Graphics Book.
If you wondered what it said on this blue label, I'm confident this it what it said:

Of course, this instruction label was designed for the turbo lifts but some passages read like it could also pass for a travel pod (and most of us didn't get a better and closer look, anyway).
Had the set decorators intended to pass these doors as something else (cover hatch for space suits or engine components) they could have theoretically used other and more appropriate label leftovers.
Possibly the set decorators thought they'd be dealing with the previous travel pod, while the VFX guys in post-production thought of the new and extended version, instead.
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