I personally don't buy either of these theories. The Air Force has no history dating back to "colonial times." The Air Force became a separate service (from the Army) in 1947. If this "custom" dated back to colonial times, it would have been infused in the Army, as well. Wherever the term "shirt" came from, it's most likely from sometime after 1947, when the Air Force separated from Army customs and influence.
One final note about the term "shirt:" Prospective first sergeants who are "shadowing" Air Force first sergeants as part of the selection process are known as "Under Shirts." Air Force NCOs who temporarily fill in for the first sergeant when the "shirt" is on leave, deployment, or TDY, are often referred to as "T-Shirt," (The "T" stands for "Temporary.")
Original Author of First Sergeant History Unknown. Updated by Rod Powers in 1999, and 2003