Putters do not swing vertically back and through along the target line. Because of lie angle, the proper putting stroke moves along an arcing path: slightly inside of the target line, back to square, then to the inside again after impact.
The proper length putter correctly sets eye position just inside the target line, and the correct amount of toe flow allows the putter to flow squarely to the proper arcing path throughout the stroke.
The best players in the world keep the face square to the arcing path and the butt of the putter pointed at the midsection throughout the stroke. Selecting the right putter will help you do the same.