David, kind of funny, you posting your 1916 LP08, as today I received one very much like yours from a friend.
Mine started life as a 1916 DWM Lange Pistol, then was reworked by the Weimar government and issued to the police. It is all matching, with matching holster, originally an LP08 holster (commonly called an Artillery holster).
According to Still in Imperial Lugers, 1991, there were 140,000 DWM 1916 pistols made, with another 20,000 made into the Long P.08.
This shows the barrel has been re-barreled at one point, as evidenced by the cut in the receiver above, for the rear sight.
This gives a good example of the differences in the P.08 and the LP08, both of the other pictures are from Jan Still's Imperial Lugers, with my 1916 on the bottom right.
Mine is 1920 Reichwehr marked, but others have said, not all were, it states in Weimar Lugers, that many "missed" this stamping.
I would imagine the barrel would have some kind of Weimar marking, mine is fairly light and a bit hard to see, but is here:
It should be located almost on top of the barrel, mine is slightly to the right.