I believe the Henry had a double or forked firing pin that would strike the rim at the same time, as your case would indicate. The priming compound is located in the rim of the case, and a double firing pin would increase the chance that a strike on the rim would land in an area where the priming compound was located.
Once the case is formed, the priming compound, in liquid form would be introduced into the case, and then the case would be rapidly spun on its long axis so the liquid would be forced into the rim via centrifugal force. The liquid would be evaporated off leaving the priming compound behind, and then the powder and bullet would be seated in the case. The first self-contained metallic cartridge (in the form of the .22 short rim fire) was invented in 1857 for the Smiith and Wesson Revolver, so the rimfire was new technology. Henry developed the cartridge for his carbine (patented in 1860) and the case was first made of copper, then later of brass and the ammo was offered commercially into the first third of the 20th century. In addition to the Henry, the Spencer Carbine was developed at approximately the same time as the Henry, and it too used a rimfire round especially developed for its carbine.
The problem with a rimfire round is its priming system. The rim has to be folded over in a manner to contain the priming compound and that rim has to be thin enough to be crushed by the firing pin when struck. This system worked well for low pressure cartridges, but with the invention of the center fire primer in 1866 which allowed for higher pressure cartridges, with the exception of the .22 rimfire, the era of rim fired cartridges was practically over.