What is the difference between a private and a commercial flight according to the US Customs and Border Protection?
According to the CBP eAPIS FAQ, this definition is in 19 CFR 122.1:


  (d) Commercial aircraft. A “commercial aircraft” is any aircraft
  transporting passengers and/or cargo for some payment or other
  consideration, including money or services rendered.
  
  [...]
  
  (h) Private aircraft. A “private aircraft” is any aircraft engaged in
  a personal or business flight to or from the U.S. which is not:
  
  (1) Carrying passengers and/or cargo for commercial purposes;
  
  (2) Leaving the U.S. carrying neither passengers nor cargo in order to
  lade passengers and/or cargo in a foreign area for commercial
  purposes; or
  
  (3) Returning to the U.S. carrying neither passengers nor cargo in
  ballast after leaving with passengers and/or cargo for commercial
  purposes;