i have looked at several flights and checked their departure times and flight times and for some of them i have checked the departure time on the airline website and the departure time on flight aware and it has been 5 or 10 minutes after the departure time on the website so i assumed it would be the take off
That’s correct. The taxi time does not show. Dbaker or Mduell could better explain , but FA picks it up as they are beginning the flight plan portion of the flight so sometimes that time is after wheels up and after they are airborne.
As covered in the Questions/Answers
Why do the departure and arrival times on a scheduled airline differ from the information on the airline’s web site?
Airlines generally display the times that the aircraft departs and arrives from a gate. FlightAware presents the time that the aircraft departed and arrived, also known as the “wheels up” and “wheels down” times. At large airports or in various circumstances, there can be a significant difference in these times due to delays on the ground.