I'm sure it is the .32-40, which probably most fairly called Winchester and Ballard, as Ballard used it first. It was actually chambered in the 1894 Winchester before the .30-30, since although they had the design for the latter in mind as its main cartridge, they weren't ready with the nickel steel a smokeless cartridge needed. It is indeed one of the same family as the .38-55, .30-30 and .25-35, but not the .38-56, which is totally different cartridge with a .504 head diameter, used in the larger 1886 Winchester.
It is actually nearly a tenth of an inch longer in the case than the others, so if you want to form the case, a counsel of perfection would be to make it from the .38-55, which will lengthen a little more, as it is narrowed, than the others. But at the pressures used, a short case is unlikely to do any harm. It was noted for fine accuracy, and one of the cartridges used by scheutzen shooters, often with the bullet separately loaded from the muzzle, who for many years achieved higher accuracy than any conventional rifle.
At moderate ranges it is a thoroughly effective deer cartridge, capable of using heavier bullets than the .30-30. I've got mine, never used on game, which in the UK is an antique if made up to 1939, and subject to no legislation whatever until you want to use it. It spent most of its life in Australia, and is a button-magazine rifle, which I trust more for accuracy than one with a magazine dovetailed into the front end.