The Turkish contracts, and I believe the Japanese contracts, were chambered for the .44 Henry rimfire.
Remember the first No.3 Models submitted for Army consideration were .44 Henry rimfire.
According to Roy Jinks, the first No.3 was .44 Henry rimfire. The Army returned it saying they wanted a centerfire. S&W changed the gun from rimfire to centerfire, "without changing chamber dimensions." Comparing a .44 Henry Flat with a .44 S&W American would bear this out. I have seen unmarked specimens of .44 S&W American sold as .44 Henry Centerfire and .44 Adams.