It all has to do with whether or not the airline is scheduled or non-scheduled.
Flag=scheduled international operations
Domestic=scheduled domestic operations
Supplemental=Non-scheduled operations (which may be domestic or international)
If you work for a scheduled carrier, and fly a leg from London to New York, it would be operated under Flag rules. If you then fuel up and fly to Chicago, it would be operated under Domestic rules. If your part time job is flying for a Part 121 charter or cargo airline, you would fly under supplemental rules.