I've been at Compass for a while. Our Out time starts on brake release. Our AOM dictates that we release the brake once we have a push clearance from ATC.

There was a huge compliance push here a few years ago because apparently there were Delta folks sitting in the tower in LAX timing the difference between when an aircraft indicated it had blocked out (which I guess is transmitted automatically to the ramp tower in LAX so they can see what gate is open) and when the same aircraft called for the push clearance. I believe they even hauled some Captains in for section 19s.

It still failed. We pretty much drop the brake when we're ready to go and everything is closed up (it helps that 99% of the time this will be indistinguishable from following the AOM guidance). My one caveat that I do is I wait for the FO to make his first attempt at contact with ATC to request a push clearance. Then I drop the brake and if ATC wants to tell us to hold for 30 minutes, that's their deal.