To clarify, what I've always heard is this:
.380, 9mm Kurz and 9 X 17mm are different labels for the same cartridge.
This cartridge is sometimes called 9mm Browning Short as well. And 9mm Corto if memory serves. It depends on which country it was loaded in.
Years ago, I owned an AMT .380 Backup, right on the gun was stamped "9mm Kurz" just after the .380.