The idea that Cotterill means cottager is a kind of myth. The trouble here is the false assumption that the name is Anglo-Saxon in origin and one of those names based on occupation. Since cotter means cottager, there you go. Well, if you don't mind being a sick cottager, anyway.
In reality the name is Norman in origin and arrived in England along with, and after, William the Conqueror's victory at the battle of Hastings. Men bearing the name Cotterill (or a variant) fought in that battle, one of them being granted lands in Derbyshire as a reward for his contribution to the favourable(from the Norman perspective) outcome. Whether a particular instance of the name goes all the way back to that moment of glory (again, taking the Norman view) is, of course, another matter!
You might be interested to know (I know I was) that a Thomas Cotterill apparently fought at the battle of Waterloo - on the English side this time; we're a flexible family. Still another Cotterill waged heroic war for the English in the Spanish theatre of that same Napoleonic War.