there are different criteria to explain pivot point.these are the situation :
1) pivot point when vessel making head way with no wind
2) pivot point when vessel making head way with wind.
3) pivot point when vessel making head way with trim forward or aft
4) pivot point when vessel going astern same with wind without wind and trim parameter
5) pivot point when vessel is stopped .

elaborate each one of them its a lengthy answer.
point point is fwd when the vsl is making headway it keep on shifting fwd as the speed increased 1/8 of the ship length. she has a good turing lever when the vessel moves fwd because of the distance in the pivot point and point of thrust. which will tend to turn the vessel. same way when the vsl is going astern pivot point shift aft . and when the vsl is stopped pivot is in the centre.
if wind is there on even keel draft the wind a always considered in the centre of the available body above the WL. hence if vsl is even keel it is in the centre and if trim by stern its is shifted lil fwd and vice versa.
the distnace of the pivot point and wind also play a role in vessel maneuvering if the vessel is down by head an making head was point of wind is in the aft becuase the vsl is down by head and pivot point is fwd because the vsl in making head way hence larger turning lever the wind will then turn the vessel towards the windward side.

correct me if i am wrong thanks.