Just a few observations:
In Season One there are turbolift call buttons (up or down) on the left side of the turbolift door, which Kirk used in the second pilot WNM. After that I have thus far not been able to see these being used.
At least these should somehow signal the waiting passenger how long it will take for the next cabin to arrive.
There are also incidents when Captain Kirk enters the cabin, pulls the handle to say where he wants to go and releases the handle while turning around. I presume pushing (go down) or pulling (go up) the handle is also some sort of manual instruction / override. In EI Spocks tells the turbolift to get to "deck 2" to bring the Romulan Commander to her quarters. I believe he was pushing the handle down to let the ship's computer know he was referring to "engineering deck 2".