Let’s just leave out the words “moves under its own power”, because they are inconvenient.
All,
This does raise the interesting point that most airline aircraft automatic time stamping is triggered by the initial "brakes off" signal, terminated by the brakes on signal, from the position of the parking brake handle/lever.
At least one operator of my acquaintance has this anomaly covered by a legal instrument, so the time being moved by the tug is counted, and the crews can't be done over for logging false time, as they almost all use printouts from the company records for their "log book".
Tootle pip!!