Using a conventional (as opposed to a long) putter there are essentially 2 ways to make the putting stroke.
The first way is to take the putter back and inside the line. This involves turning (rotating) the shoulders to make the stroke. Because your spine is at an angle the putter head has to move inside the line on the back swing and back to the line on the forward swing. Geometrically, the closer to horizontal your spine is when making this stroke the less the putter head will move inside the line. But there are limits as to how horizontal you can make your upper spine in your set up.
Using this method the club face opens during the back swing and returns to neutral when it gets back to the ball.
The second way of making the putting stroke is to lift your right shoulder and drop your left when taking the putter head away from the ball. It is essentially a rocking motion. In this way you can take the putter head back and away from the ball following the line and return the putter head to the ball along the same line. The face of the putter remains at 90 degrees to the line throughout the stroke.
I consider myself a fairly good putter and I use the second method. Contrary to most opinion I allow a small amount of hip and upper body movement during the stroke which smooths things out a lot.
I am toying with idea of having a putter made with a longer shaft and and a very upright lie angle. This will enable me to stand very upright and almost eliminate shoulder turn in the putt. I will update here on progress with this idea.
Experiment with both methods in your putting stroke and go with what feels most comfortable and has the best result.