I would not include the “Gew. 88” in the designation of these rounds, myself. While the bullet shown is round nose, it is a soft-nose and not specifically for rifles of the Model 1888 action. Many German Mauser-action sporting rifles were made with the .318 bore for years after the “S-bore” (.323") appeared on the military rifles. I have often wondered in fact, whether or not it was required by German law that sporting arms use the .318" bore of the pre-1905 military rifles.
I have in my own collection the lacquered-steel cased rounds with the headstamp shown with four different bullet types, and with copper-washed steel case, two more different bullet types. One of the rounds in a lacquered steel case has the steel-cupped primer. I suspect these round were made over the entire span of the WWII years, and perhaps even slightly before. Rheinisch- Westf