Thanks for the question!
You are likely opening the rear armpit a bit much in the backstroke while also flexing the lead wrist in sort of a "hooding" action from olden days, a manipulation designed to keep the putter face aim the same as address aim or parallel to it. This is usually associated with Horton Smith, 1934 Masters Champ. If you set up the putter face aimed down a yardstick and then moved it straight back, it will seem to involve the lead hand cupping a bit to keep the putter face aimed dead down the stick. Actually, that is not necessary if the shoulder action is in a vertical plane: then the putter face moves straight and the face does not require manipulation to keep aimed straight.
But what should you be doing and experiencing?
If you set up with the shoulders aiming the chest parallel to the putter face aim, and the pivot of the chest (either the base of the neck or a LINE or PLANE sticking out of the middle of the chest the same orientation as the leading edge of the putter head), that will also align the shaft to the midline of the body. Then the stroke is basically like the rod and pendulum bob of a grandfather clock:
The arms and hands and putter make one shape that does not change, and this WHOLE is the pendulum that swings beneath the pivot point off the base of the neck, midway between the shoulders. The whole shoulder frame works up-down like the wings of an airplane making a banked turn, but in a banked turn the nose of the plane stays aimed straight until the wing rock takes effect and curls the nose around. A pendulum putting stroke is just like a plane rocking the wings into a bank but before the nose changes direction.
So if you set up a putter face aimed perpendicularly into a wall, and then set up the chest so that each shoulder was the same distance back from the wall, the shaft will line up with the midline of the chest. The BUTT may or may not point at a specific place along this line, but it will point somewhere along it. Then when the shoulders rock, the WHOLE THING swings together beneath the base of the neck. So the WHOLE THING aims at the base of the neck. For some folks that includes the butt of the club.
If at address the butt of the club aims at the base of the neck, then the lines of your forearms also aim at the base of the neck. For many people, the way their body is and the way they hold their putter and the way their putter is designed and the way they set the putter down in relation to the middle of the body results in lots of different aims of the butt of the putter at address and during the stroke.
My advice is to set up next to / across from a wall, and then keep the chest oriented the same as the wall when you rock the shoulders, and keep the base of the neck steady in one place. If there were a pencil poking out of the base of your neck parallel to the floor aiming perpendicularly at the wall, and you made a "perfect" shoulder stroke, the pencil would simply spin in place but would not change aim or swing left or right, or bob up or down either -- like the nose of a plane starting a bank turn. That sort of setup and stroke move will keep the shaft aiming into the midline of the body the same way the rod of a pendulum aims at the hook in the cabinet it swings from.
In addition, you don't really need a perfect pendulum or shoulder stroke anyway. ALL strokes between the feet do not involve opening of the armpits. The definition of a "shoulder stroke" is a movement that keeps the armpits from changing angles in the stroke (and nothing changing in the arms or hands either). The back armpit often opens a little but only after the putter head clears the rear foot. In the forward stroke, the lead armpit opens a little once the putter head clears the front foot. But IN BETWEEN the feet, especially coming into impact, the armpits don't open for anyone except the worst weirdos. This means that EVERYONE uses a really good shoulder stroke action in the forward stroke between the feet. All you have to do is get the putter face moving into the stance online and square and then not twist the chest while this forward stroke is under way.
As noted, some putter designs and some grip forms shift the bottom of the putter to the side of the middle of the body. For example, a left-hand-low grip moves the bottom of the putter to the front side of the midline of the body, perhaps to the heart or shirt pocket. Likewise, an offset hosel or plumber's neck hosel results in placing the putter down to the front of the midline also. In these cases, the shaft will usually aim to a plane sticking perpendicularly out of the chest that splits the heart line left of the chest's midline. Those combinations usually have the handle aiming at the left pocket or heart or something close to that.
Putting Coach and Theorist