How does ICAO differentiate between Commercial and Private flights?
According to ICAO Annex 6 Part 1 (PDF):


  The Standards and Recommended Practices contained in Annex 6, Part I,
  shall be applicable to the operation of aeroplanes by operators
  authorized to conduct international commercial air transport
  operations.


According to ICAO Annex 6 Part 2 (PDF):


  International general aviation operations with:
  
  a) aeroplanes with a maximum certificated take-off mass exceeding
  5,700 kg; or
  
  b) aeroplanes equipped with one or more turbojets engines.




The International Flight Plan form also requires flight type to be declared, but I am not sure that this would be ICAO's sole requirement.

Type of flight—Indicate which one:

S—Scheduled air transport
N—Non-scheduled air transport
G—General aviation
M—Military
X—Other




For a simpler explanation, this PDF lists civil aviation activities:





This page contains a table (at the bottom) classifying each type.