"Part of my assumption is that the Bergmann, Steyr, and Browning Short (not to mention .38 ACP) cartridges were around, though I've no idea if they made an impact on the American market at the time."
I can answer that...
No.
The 9mm Browning Short, Corto, Kurz, whatever, was sold in the United States as the .380 ACP/CAPH. FN and Colt had an agreement -- FN got Europe, Colt got the United States, and the two wouldn't cross over.
Some 9mm Steyr pistols may have made it into the United States was war trophies, but it would have been an extremely limited number.
The 9mm Bergman didn't make it into the United States in any quantity until after World War II.
Same with the 9mm Mauser.
Same with the 9mm Browning Long.
While there were lots of 9mm cartridges in production world wide at various times in the first few decades of the 20th century -- 9mm Japanese revolver, 9mm Dutch revolver, 9mm Glisinti, 9mm Mars, various earlier black powder pinfire and patent ignition cartridges, plus the ones listed above, the ONLY one that was in production in the United States was the 9mm Parabellum.
American cartridges of nominally 9mm, including the .38 S&W, the .38 Short and Long Colts, the .38 Special, the .357 Magnum, the .38 ACP, the .38 Super, etc., were NEVER marketed or otherwise identified in the United States as 9mm cartridges until well after World War II, when European and Asian 9mms finally started to flow into this country in significant numbers as military surplus. And then it was a case of "Oh, in Europe my .38 Special would be a 9x28R" or some such. It was an interesting tidbit, not a focus for confusion.
Even with all of the surplus coming in, the only way to fire most of these guns was to obtain surplus military ammo, foreign-made commercial ammo (IF it was made), or go the George Nonte cartridge conversion route because American companies never made ammo for these guns.
That's the long way of saying that no avenue of confusion existed.
The 9mm Luger designation was, I'm certain, chosen to identify that ammunition with the guns that Stoeger was importing.
Nothing else.
And no, there never has been a uniform system of nomenclature.
Anywhere.
Probably the most universal system is the European metric system and the old American blackpowder systems.
That's because you had private companies doing the lion's share of cartridge development both in the United States and in Europe, and cartridges were named primarily out of marketing considerations.
It's analogous to automobiles, really. Each company picked the name it wanted.