Why is the conserved Lagrangian energy $E$ equal to the total energy in this example but not in a similar example?

Question

I am aware that there exists duplicates to the title and have gone through the answers but it still doesn't answer my issue with a statement in the last image.

These two similar situations with slight variations are being introduced, I am not able to understand the reasoning provided as to why the quantity $E$ is equal to total energy in one and not in the other.

I am not able to understand how the minus sign before $r^2ω^2$ implies that the conserved quantity is not total energy as stated in the last slide.