Your meter does not really know where ground is. It only reads the voltage across the probes. So when the meter reports L1 to Ground = 0V It only tells you there is no voltage difference. I think L1 does not equal 0V but ground is at 110V.

Is there a ground from a different circuit nearby you can get to. (from a 110v plugin) Use a copper water pipe as ground. The point is I want a meter reading from a known ground and not that outlet. I know I said this several time but your ground is not connected or poorly connected.

When I used a copper water pipe as ground for the meter, I still get the same readings:
L1 120v L2 120.
When I hit the start button, L1 goes to 0V and L2 goes to 240 V

 

When I hit the start button, L1 goes to 0V and L2 goes to 240 V

I do not know what to do now.
A broken L1 can loose voltage, but L2 doubling in voltage... ??
Good luck.