FWIW, the spelling is Luger. Just that. Not Lugar, not Lüger, not Leuger, not Looger, not Lugger. It was the name of the man who modified the large and somewhat awkward Borchardt into the handier pistol the Germans called the Parabellum pistol, from the trade name used by DWM. It was called "Luger" in this country after Stoeger became the importer and trade-marked the name.
The term "Pistole 1908" or P.08, was the German Army designation for a 9mm pistol. It does not apply to a .30 Luger or to any Parabellum-type pistol not made for the German Army.
There is an interesting sidelight to the story of the 9mm Parabellum. When the German army expressed a desire for a larger caliber, they specifically mentioned 9mm. Luger, for obvious reasons, wanted to keep the same base as the .30 caliber (7.65 Parabellum) so he first tried using a case with a small shoulder. But that didn't give enough case support, so he decided to support the case on the case mouth. That resulted in a tapered case, something that has not always been beneficial with long columns in a straight magazine, like SMG magazines.
Meantime, Browning started with revolver cartridges and tried to reduce the rim for better feeding, ending up with semi-rimmed cases. But at some point, he saw the 9mm Parabellum case and recognized the advantage of supporting a case on its mouth. He went one better, though, and made his cases straight from that point on, so the .380 and .45 ACP cases are straight and feed much better than the .32 ACP, 9mm Browning Long, and .38 ACP.