Usually when a flip coil overheats it's because the pull winding is left powered by a problematic EOS switch. In your first pic, the two opposite switch blades were shorting against each other that could have supplied power to the pull winding erroneously.
hot flipper coil lwr (resized).jpg
In the second pic the coil is only lightly toasted, not burned. If left unattended, an overheating flip coil will char the wrapper black. I assume the 2nd switch on the 2nd flipper feeds the 5th?