The inner / internal oblique abdominal connects at one end up and down the side of the rib cage and at the other onto the pelvis, so it binds the upper skeletal structure and the lower.
Contraction tugs the rib cage sideways down at the pelvis, as in a side crunch exercise.
If you stood erect with hands straight above the head in a "hands up" posture and then did a side bend to the left, the hands will point from 12 o'clock to 11 o'clock. This "bends" the spine laterally but does not rotate it.
If, however, you rotate the spine, that movement is done by the lower back muscles tugging the bottom of the rib cage, as in the right quadratus tugging the right bottom of the rib cage to the rear clockwise (seen from the head looking at the base of the spine).
This action, performed with the "hands up" posture, turns the hands from 12 o'clock to 1 o'clock rotationally above the feet but there is not much bending laterally.
Call the first move A and the second B.
Bending the upper torso forward as in a putting stance, but with the "hands up" starting posture with palms facing straight ahead, and then performing move A with the inner oblique tugging the left shoulder and side of rib cage at the left pelvis, the hands "bend" to 11 o'clock but the palms don't rotate much at all (perhaps a smidgen). The line across the hands and the shoulders stays aligned with the hips and feet.
But performing move B moves the hands to 1 o'clock and also twists the palms and chest and shoulders to aiming a bit diagonally off where the chest aimed initially, now aiming to the right of that -- the line of the shoulders and hands now athwart the line of the hips and feet.
So both of the moves you describe fail to capture move A. Move A keeps the putter face square and avoids the spine twist that rotates the shoulders and chest out of the line of aim at address.
If you think of the line of aim on the ground as being a vertical wall along that line, the chest in move A starts and stays parallel to the wall, but move B turns the chest away from this wall.
So set up opposite a wall in the house and "start the backstroke with a shove of the lead shoulder" (performed by contracting the lead-side inner oblique abdominal).
Putting Coach and Theorist