The Henry wasn’t used that much, it was quickly replaced by the Winchester. But yes, most of these carbines used a pistol class cartidge, such as the .44-40, etc.
From Wiki: *“The Henry repeating rifle was an American .44 caliber lever-action breech-loading rifle designed by Benjamin Tyler Henry in the late 1850s. The Henry rifle was an improved version of the earlier Volcanic Repeating rifle. The Henry rifle used copper (later brass) rimfire cartridges containing 25 grains (1.6 g) of gunpowder to a 216 grain (14 g) bullet. 900 Henry rifles were manufactured between summer and October 1862; by 1864, production had peaked at 290 per month. By the time production ended in 1866, approximately 14,000 units had been manufactured.”
"The original Winchester rifle was famous for its rugged construction and lever-action mechanism that allowed the rifleman to fire a number of shots before having to reload: hence the term, “repeating rifle.”
The first model, the Model 1866, was nicknamed Yellow Boy because of its brass receiver. The Model 1873 was Winchester’s next design. The 73 was much more popular than the 66 because of the iron (and later steel) frame which allowed it to take the newly designed and more potent centerfire .44 WCF (Winchester Center Fire, also known as .44-40) cartridge. The 1873 is often called The Gun That Won The West.
The 1866 was only available in the rimfire .44 Henry. The 73 was available in .44 WCF (.44-40), .38 WCF (.38-40), and .32 WCF (.32-20), most of which were also available in Colt, Remington, Smith & Wesson, Merwin & Hulbert, and other revolvers. Having a common centerfire cartridge in both revolvers and rifles allowed the owner to carry two firearms, but only one type of ammunition. The original 73 was never offered in the military standard .45 Colt cartridge; only modern reproductions are offered in that caliber. …
Winchester continued to dominate the American rifle market for decades with the introduction of Models 1876, 1886, 1892, 1894, and 1895 (which featured a box magazine, rather than the tubular magazine found on the previous models). The '76 was a heavier-framed rifle than the '66 or '73, and was the first to be chambered for full-powered centerfire rifle cartridges, as opposed to rimfire cartridges or handgun-sized centerfire rounds. …
…the majority of lever action rifles seen in classic Hollywood Westerns are Winchester '92 carbines chambered in .44-40 and .38-40 (to utilize the “5-in-1” blank cartridge), which John Wayne famously carried around through dozens of films set in periods from the 1830s to the 1880s."*
They made millions of Winchesters, likely more than 1000x as many Wichesters as Henrys. A few Henrys were used in the Civil war, giving them some claim to fame.