"Based on illustrations and archaeological finds, we may assume that medieval awls were most likely stabbing awls."
It does continue by saying of diamond awls "there is some evidence that it was also quite common throughout the Middle Ages as well." from: http://www.personal.utulsa.edu/~marc-carlson/shoe/RESEARCH/GLOSSARY/bdefa.htm
Do you have more evidence from somewhere? The above statement weights the round cross section stabbing awl slightly more than the diamond awl.
Awls for useages other than shoemaking may differ, I haven't read even basic stuff about them.
Honestly interested - what are your sources? Tiff