Part of the problem with the original No. 56 cartridge (aka .56-56, which was the dimensions of the case just in front of the rim and at the mouth) was that it utilized a heel bullet, lubricated outside the case, and was approximately .54 caliber. (Actual bullet diameter outside the case varied quite a bit from manufacturer to manufacturer! By reducing the bullet diameter and the bore/groove diameter, Springfield Armory was able to use an inside (the case) lubed bullet, with a substantial crimp by the case mouth on the bullet, insuring the bullet would not be jostled loose and the lube wasn't likely to pick up dirt, being protected by the case. Although called the .56-50, the barrel dimensions were: bore .50 caliber, groove diameter approximately .515". Christopher Spencer thought the Springfield cartridge had too much crimp, and so designed a more bottlenecked case, but used the same bullet diameter. He called this the .56-52. In reality, the .56-52 would chamber and shoot just fine in the .56-50 carbines. Since these were all rimfire cartridges, they wouldn't be reloaded, and the .56-52 case fireforming to the chamber was no problem. As a matter of fact, all three cartridges would shoot in the older M1860 Spencers, if one could accept a little less accuracy. The .50 caliber bullets would generally upset to fit the larger M1860 Spencer barrels, although accuracy probably wasn't as good! The Spencer sometimes had a reputation for excessive drift. One wonders if this wasn't due to shooting the .56-50's in the .56-56 chambered guns? Army logistics being what it was in the post-CW Indian Wars campaigns, the newer ammo might have been issued to units that still had the M1860 carbines and also the M1860 rifles (3rd Infantry being the only doggie outfit West of the Mississippi armed with repeating rifles).
If you look at the sequence of cartridge development, the next step for Springfield was to lengthen the case, and change to a central fire design, but using the internally primed case design. This was, of course, too long to fit in the Spencer, but worked well in the Allyn-conversion breechloading Springfield .50-70 rifles, and subsequent models, including the Sharps conversions.