Here is the progression from the .44 Henry to the .44 Magnum:
According to Roy Jinks, the first S&W No. 3 submitted to the Army was a .44 rimfire. The Army wanted a centerfire revolver, so S&W "changed the gun to centerfire without any changes in cartridge dimensions." So the .44 S&W American is a .44 Henry centerfire, for all practical purposes. Some Winchester Yellowboy Model 66 were made for the "Henry centerfire" round.
From the American came the .44 Russian. Incidentally, the first .44 Russian cartridges were outside lubricated, inside lubrication not being incorporated until about 1880 or so, when introduced by the Union Metallic Cartridge Co.
No the rim fire .44 caliber round is long gone. The old rim fire cartridges were not very dependable. They would often misfire due to uneven fulminate priming in the rim. The shooter would have to turn the cartridge around and snap the hammer again. The double firing pin evened the odds.
With the advent of the Berdan and Boxer primers the rim fires were soon replaced by the center fire. The .22 rimfire the oldest of the rim fires dating to the 1850s is still with us.
The rim fire case was essentially nothing more than an extended percussion cap. They were loaded first for the S&W Modl. #1 revolver in .22 shorts or 5 MM bullets. These .22 shorts were little more than a percussion number 11 cap loaded with a bullet.
The .44 Cal. 11 MM and many other large caliber rim fires would be short lived systems.