What we normally know as the 9mm AND also the 9mm Luger, 9mm Parabellum, 9mm P, 9mm Nato and 9x19mm are the same round and they are all names for the standard cartridge all commercial semi-auto handguns SMGs and carbines are chambered for.
The 9mm Automatic (or 9mm Auto) was a prototype round that to my knowledge no one has taken on commercially. As you noted it has a straight case, as opposed to the very mild taper the original round has. This straight case is a “feature” of dubious value to this particular round which has been around and working effectively for over 100 years in it’s standard form.
Your 365XL is chambered for standard commercial 9mm ammunition and I can basically guarantee you it will run flawlessly with any quality commercial ammo off the shelf. The 365s I’ve fired have been absolutely reliable and above average accurate as just about all SIGs tend to be.