You are pretty close. If my math is right then if you convert .357 inches to millimeters you get
.357 inches = 9.0678 millimeters
And the 9x19 parabellum round is actually 9.017 mm or .355 inches.
And for some crazy reason the .380 is actually .355 inches...
go figure.
Some rounds measure by casing and some by bullet and some just make no logical sense but have their foundation in how they relate to some other round or its history of development. For example the 30-30 uses a .308 bullet, but the second 30 came from the early loads of 30 grains of powder...and the 444 Marlin’s name is due to it being a derivative case of the 44 magnum with .429 bullets and so on it goes...
The 9.0678 mm value is actually based on a legally non-existant unit that would be an “international inch.” (ie 25.40 mm = 1 int inch)
But the real inch is 12th of a U.S. Survey Foot (the only legally useable foot), which is 0.30480061 meters, thus 25.400051 mm = 1 inch, so .357 = 9.067818 mm.
Let’s add to the confusion:
“Rifled bores may be described by the bore diameter (the diameter across the lands or high points in the rifling), or by groove diameter (the diameter across the grooves or low points in the rifling). Differences in naming conventions for cartridges can cause confusion; for example, the projectiles of the .303 British are actually slightly larger in diameter than the projectiles of the .308 Winchester, because the “.303” refers to the bore diameter in inches, while the “.308” refers to the groove diameter in inches (7.70 mm and 7.82 mm, respectively).”