He wasn't grasping at straws at all, he was just being accurate with the terminology. There's a major difference between a weak battery and a dead battery where jump-starting is concerned.
You would be wrong on not being able to push start an automatic equipped vehicle too. While they have to be pushed a little faster, you can definitely still push-start many vehicles that have automatic... the automatic's torque converter couples the engine to the drivetrain in both directions. I push-started my '57 Chevy with its Powerglide automatic at least once during my high school days.
But if the battery is dead like if you left your lights on, you're not going to be able to push-start it no matter what transmission you have.
If I remember right we had to get those 57 Powerglides to about 30 mph but they very well could be push started