To me, playing fast means running and moving the ball upcourt as quickly as possible, a la the Nuggets or the Warriors. The shot selection is definitely part of that- That would be run and gun. Push the ball, by contrast, to me means move it upcourt quickly, but if there is no break available, look for a good shot. In both cases, it is my humble opinion that the increased pace with which the ball is moved upcourt will lead to more turnovers than if the PG were to run back to whomever rebounded the ball, take it out of his hands, and walk the ball upcourt- then move the ball around the perimeter looking for an open big man or wide open mid-range jumper- be patient or look inside.
Once I scored a basket that still makes me laugh.