This may vary from airline to airline, but I'd be surprised if the major airlines' websites don't all take departure/arrival data from their ops systems.

I remember talking to my partner once when she was on board a delayed BA flight. I was refreshing the ba.com screen from time to time to see whether the ops system had updated the departure time. Suddenly she said, I have to get off the phone now because we're just pushing back. A refresh of the screen showed an immediate change from "estimated" to "actual" departure time.