This seems like a good way to look at things.
I think a better way to classify them is
small game calibers
American big game calibers
African big game calibers
That's essentially what the old European system was designating.
With black powder as the most common propellant, and a major limitation to cartridge performance, all that really mattered was "how big and how heavy".
Bullet diameter and weight determined what the cartridge was good for.
Later on, you had terms like "Express" added to some cartridges, to designate light bullets at higher velocity (rather than the "standard" for the cartridge).
With the introduction of Cordite and Smokeless powder and expanding jacketed bullets, most of the small/medium/large-bore references fell apart. It was a new era, where each cartridge could be judged on its own merits, rather than just bullet diameter and weight.