Not exactly.

I often take a photo at the point of wheels up (and down.) The departure time I see on my last 4 flights is the time we pushed back from the gate.

One flight showed departure time as 3:00pm. I have photos from the taxiway at that time and wheels up was only 3:18 pm.

Another recent flight showed departure as 6:56 am. Um, no. That is the exact time we pushed back at at 7:10 am, we were still taxiing away from the gate. At 7:17 am de-icing started and we were in the air at about 7:30 am.

And on another recent flight, FlightAware shows departure as 11:58am, but wheels up was at 3:17 pm
Interestingly, on that flight, arrival showed 4:47pm and that is when we pulled up to the gate.

These are just small and few examples.

I actually found the departure and arrival times on FR24 more accurate that FlightAware for some of my flights but then I didn't design the algorithms that track all of this, but the time stamp on my photos is the accurate reflection of where my aircraft was at what time.