Yep, these guys are right. The most ubiquitous "cowboy" round would probably be the .45, specifically the .45 Colt, which was adopted for their Single Action Army but ended up being used by the Military for close to 20 years and is still a popular cartridge today. The .45 was pretty much the standard at the time, though as others have stated there were plenty of revolvers chambered in .44, .39, .41 (check out the Volcanic pistol for a cool little example of this caliber), .31, you name it.
Others have said it already, with a little bit of variance on the date, but the .357 Magnum (technically the .357 S&W Magnum to differentiate from the .357 SIG round) was introduced in 1934 to replace the .38 Special, its parent case. There is no "non-magnum" version of the .357 cartridge, and cowboys would obviously not have had access to them.