Your gun is a Model 1905 "Winchester Model" manufactured ca. late 1911-early 1912. While it is mechanically identical to the .38 Special "Military & Police Model" it is a separate model.
There is no reason to not use "modern" .32-20 ammunition because there is no such thing! All currently manufactured .32-20 is loaded so as to be usable in any firearm chambered for it, revolver or rifle. At one time there was a load intended for the Model 1892 Winchester or equivalent Marlin rifles, but it has not been made for over 50 years. It has a round-nose jacketed hollow-point bullet, where all other .32-20 ammunition has flat-pointed bullets. Anything with flat-point bullets is OK as long as the revolver is in safe mechanical condition.