I also read the info on that site. First, the actual case is small ring, large shank, not the reverse. Those rifles came about c. 1908 when the Germans decided to produce and adopt as standard a lighter and shorter version of the long Gew. 98, but not as short as the cavalry carbine. That short rifle was designated the Kar. 98AZ. One step taken to lighten it was the use of the small receiver ring, but with the standard barrel shank. It was later designated the Kar. 98a. (Please don't ask me to type what the "AZ" stood for.)
Germany produced several million Kar.98a rifles before the end of WWI. In their work with the short rifle, though, they found it had problems, in part because of that small receiver ring, so they abandoned that experiment and later adopted a different "carbine", the K.98k. So Germany had a gazillion rifles they didn't want and couldn't use with the small army they were permitted; Turkey, dreaming of restoring its power in the Middle East, wanted more and more rifles. The solution was obvious and the Kar. 98a rifles went off to Turkey.
In addition to those Kar. 98a's, the Turks also had Model 1890 and 1893 rifles (the old small-ring Mauser with the single column magazine); the Model 1903, pretty much a standard Gew. 98; the Model 1905, a carbine version of the Model 1903; and the Model 98/22 from Czechoslovakia, a standard Brno long rifle.
In the mid-1930's the Turks decided to take two steps to modernize their small arms. They considered the Model 1903 their standard,, so they refurbished and upgraded all those old rifles to give as much as possible the appearance of that model. At the same time, they adopted the 7.9 (8x57JS) as their standard caliber, replacing the old 7.65x54. Most information seems to indicate that the Turks did use the term "Model 1938" for all those upgraded rifles, though I have seen no absolute proof of that.
As best as I can determine, the Turks never made Mauser receivers; had they wanted to do so in the mid-1930's the machinery and tooling would probably not have been available, since every European country was in the process of making all the rifles it could for its own army. Turkey did apparently have barrel-making machinery and could manufacture small parts.
Jim