Small bores are .30 caliber and under. I'd include the .308"-.312" diameters as ".30 caliber" but I'd also include the 7.92x57 Mauser round as it was a .30-06 class battle rifle round as well.
The .30 caliber distinction is based in large on the NRA "small bore" rifle target being used for matches where .30 caliber rifles were common.
.375" is where "big bore" starts, that's also dependent on the cartridge. For example, the .30-30 is a small bore round, while the .380" diameter .38-55 and the .375" diameter .375 Win are both medium bore rounds. They are based on the same parent cartridge, they just use a larger diameter bullet and are "medium bore" rounds.
The .35 Whelen is another "medium bore" round. It's based on the .30-06 case, just necked up to .358". I would also classify the .375 Whelen (.375-06) as a medium bore round. It's a substantial round, but it launches a 275 gr bullet at 2,475 fps and a 300 gr bullet at 2,110 fps compared to the .375 H&H mag that launches a 235 gr bullet at 2,895 fps, a 300 grain bullet at 2,645 fps and a 350 gr bullet at 2,325 fps.
IMHO, the floor for "big bore" rounds is the .375 H&H Magnum.