Hendere - the 7.63 Mannlicher IS a rimless cartridge. All of your measurements fit in nicely with the 7.63mm Mannlicher Model 1905 (DWM 466; GR 716, 7.65 Mannlicher Model 1901, etc.). The headstamp, which is Argentine, also fits quite nicely. I suspect it is just a sloppy headstamp job. I don’t know why the rim would be turned down, since it is at the correct measurement for the cartridge caliber that the headstamp would indicate.
In the U.S. anyway, the 1947-dated cartridge from the Borghi plant in Argentina is the most common 7.63m/m Mannlicher cartridge encountered. It was sold here, along with Model 1905 Mannlicher pistols, by the hundreds of thousands of rounds about 35 or so years ago, and still shows up in box-fulls occasionally.
I’m sure you know it, but in case someone doesn’t, the full name to fit the headstamp is F