Yes, 9x19, 9mm Luger, 9mm Parabellum are all the same thing. 9mm NATO is also the same case and bullet but is an higher pressure variant.
I beleive the actual designer named it 9x19 Parabellum.
NRA Lifetime Member Since 1999
"I ask, sir, what is the militia? It is the whole people except for a few public officials." George Mason