Joseph Stalin and the Bomb

Yes that is true.

As @LSLGuy pointed out (a decade plus ago :exploding_head:) what exactly Petrov did is not entirely clear and even whether he refused to tell his superior. As has been pointed out, the feed from the early warning satellites went to other places besides his command center.

Once the computer classifies the signal as an indicator of a launch, it generates an alarm that is automatically sent up the chain of command (especially if it is not a single launch). There were at least three assessment and decision-making layers above the command center of the army that operated the satellites - command centers of the early-warning army, the Air Defense Forces (which was a separate service back then), and, finally, the General Staff.