All M1908 Mannlicher Schönauer rifles and stutzen were originally chambered at the Steyr factory for 8X56 MS, just as all M1903, M1905, M1908 and M1910 were cartridge specific.

Here are Stoeger's MS offerings for 1927; only the 'proprietary' chamberings of 6.5X54 (M1903), 8X56 (M1908), 9X56 (M1905), and 9.5X57 (M1910):

By 1931, the 'U.S. Model of 1906' (.30-'06) was added with its longer action and magazine. These had originally appeared stateside as the 'M1924 Sequoia', made for the Sequoia Import Company before Stoeger became sole importer of Steyr arms to the U.S.:

By 1939 the MS was available in 7X57 and in 7X64 as shown in the Stoeger catalog, but Stoeger did not list them by their M---- designations at that time, only as 'carbine' and 'High Velocity'. The 'High Velocity' would correspond to the M1924 and 'M1925' MS with their longer actions than the previous models.

Here, from a British retail catalog of 1928, a '7mm' is listed with 'High Power' offerings, as set apart from the proprietary models. It does not specify, however, if that is 7X57 or 7X64.

I suspect the pre M1950 7X57 were built on the action of the 'M1925' as that was essentially an M1924 with a different magazine, chamber cut, and barrel. The M1924 and later models had a guide ring added to the magazine at the cartridge shoulder which allowed for use of different cartridges in the same model without as much expensive retooling and machining for each, whereas the M1903, 05, 08, 10 had the cartridge, magazine follower and magazine well areas designed and machined as a system, each model specific to one proprietary cartridge. The original M1924 'Sequoia' models were all .30-'06, the 'M1925' and later models were available in several different chamberings.

The .308 was available from Steyr by 1962 in the 'Model 1961 MCA' or 'Model 1960 MC' configurations: