You listed two extremes to demonstrate the difference, which is fine, but what about the example of Trump encouraging insurrection? It seems just a teeny bit relevant here, given that it is what prompted the case in the first place. And insurrection is definitely related to governing, so you can't discard it from consideration just because it's nothing like murdering a business partner. He could claim that an illegal decision was about to be made and so he had to use his executive authority to counteract it. That sounds like an official duty to me.