What is the purpose of knowing this?

Unfortunately each site may very well be correct. The problem is they are using different events to mark "departure" time. AA will use expected door close time. Others will use estimated start of taxi time. And yet others will use wheels up time (not sure if that is wheels off the ground or gear up and stowed).

The really important one to worry about is AA's originally scheduled departure time as that is what has been used to determine if you made it to the gate in time. For heavily loaded flights you should pad this a fair amount.