Maybe this experiment will help. I'm using a manual pivot in the example, the visuals are ETC with a right pivot. Use a cue ball that has a spot on it or put a chalk mark on the cue ball. Set up a straight in shot. After you get the visual rotate the cue ball until the spot is on the 'equator' and it marks the place where you place the cue tip pre-pivot. Be exact as possible. The mark should be 'just' visible on the left side of the cue ball.
With everything set up, get your ETC visual. Without moving your body tap the object ball a little bit with your cue stick to set up a slight cut to the right. To get the same visual on the cut shot you'll have to twist your body counter clockwise a bit. As you twist to obtain the visual, the spot on the edge of the cue ball 'rotates' toward the back of the cue ball and disappears. Depending on how many degrees you have to rotate your body, the spot will rotate the same relative amount. This presents a new spot (or angle) to place your tip pre-pivot. Now the same 90* pivot will give a different cut angle on the shot.
I think I'm picturing what you are suggesting. How have the table and pocket told the balls that the shot has changed?
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