Not knowing your skimmer setup (venturi vs. draw), it wouldn't hurt to leave the pump on. At least then, the filtration effect of the pump would help to keep contaminants out of the water which will need to be offset by chlorine. On our venturi setup, the return pressure drives the skim function, so it could be entirely filled with debris and won't negatively affect pump/filtration - it just would stop skimming. When storms crop up and dump tree debris, I start raking; every bit I can remove by hand is that much less that won't get sucked into the filters and increase my pressures, necessitating a filter cleaning. I check my pump cage (basket) daily and ensure there's nothing that would impede flow, along with the skimmer basket.
As to the electrical issue of thunderstorms and lightning, the grounded pump and fully earthed equipment and pool would be highly unlikely to see a strike. Running vs. not-running electrical equipment has no bearing on the likelihood of a strike either, so don't overthink it. Whatever you do, you'll want to check chlorine levels after the storm is over and adjust accordingly.