Everyone has a different idea of how to define them, including me.
My own opinion:
up to .20 = Small caliber (or "sub-caliber" - even though it's improper use of the term)
.22 to .257 = Small bore
6.5mm to .375 = Medium bore
.40 and up = Big bore
But, you may notice that most "official" references to the term 'small bore' are almost universally citing .22 rimfire rifles. They aren't necessarily defining the term, but setting standards or rules for a type of competition or event.
Likewise, most references to 'big bore' are almost universally citing .40+ bores, but not really setting a hard limit; and they generally don't include cartridges like .44-40.
"Such is the strange way that man works -- first he virtually destroys a species and then does everything in his power to restore it."