Your question prompts me to reflect further on my own remark that nobody thought in particularly technical terms. For sure, there was no book of "Best Practice" or "Risk Assessment" or even "Passage Plan". Such things were unknown until fairly recently. All knowledge was handed down by word of mouth and common sense.
This leads me now to try to explain more accurately in writing what actually did happen in the manoeuvre mentioned above. The starting point would be the Master's report, "We are ready." The ship is far longer than the relatively small stage alongside which she lies. Head-ropes and stern-ropes are on dolphins. Springs are on bollards on the stage. If the pilot is quite certain that the tugs were also in attendance he advises, "Single up to one and one each end and make fast the stern tug."
When all of that was done the next order would be, "Let go all aft", at which point the stern was bound to shift (by force of tide alone) and the next order would be "Let go the head-rope" (which is now doing nothing). The spring would be held onto for not much more than a few moments, by which time the swing had started and there was no power on the planet which could stop it. (I speak here of a Shell M Class VLCC).
The stern tug would be doing its stuff, standing into mid-stream - and the pusher on the port shoulder would be ordered to start to push as soon as the bow was coming clear and it was practicable to do so.
By those general principles, there should not have been any hairy moments on the foc'sle head.
Many thanks for the invitation to write it down after forty-odd years!!