The 9mms as we know them today are called many names depending where in the World you are living. The 9x19, 9mm luger, 9mm Nato and 9mm Parabellum are all the same. The 9mm short and 9mm Kurz are the same thing also, but we Americans call that cartridge .380 Auto. I know nothing about the 9x17, 9x18 or the Largo.
In 1908 the German Army adopted the 9mm Pistole Parabellum, ie the P08, the Luger, the German Luger pistol. This pistol was originally chambered for the 7.65mm bullet and was called the OP00. I have a few WWI and two WWII P08s from various Manufactors. Yes, I love my Lugers.