The nicknames were passed down through the generations, so I thought that the descendants of David and their families could possibly have been dubbed "Davy" to distinguish them from the other multiple Corkills around.

That is what happened with the "Dick" family from Ballajora, according to CR, where there were two main Corkill families, and one was identified by the nickname Dick. She wrote that John Corkill c.1715-1796 was nicknamed "Dick" and his son was therefore "Juan Dick". She said: "We cannot explain the nickname except to say that some distinguishing mark among so many Corkills was almost a necessity". She wrote that it could still be heard colloquially even today.
Apparently the name even carried through to those emigrating to the U.S. who often added Dick to their names thinking that was what they had been baptised.

Sue