I believe the best clarificstion there is that you have to consider the person and the office separately. When acting as the president and largely executing his duties as defined by Congress and the Constitution, he can't be charged. If the person does something outside of the office's powers then immunity doesn't hold.
Meaning, if the president shoots a random bystander on the street they can be charged with murder. If the president orders a military strike as part of an official operation and done through proper channels, they can't be charged if it later turns out the intel was bad or the strike went wrong in some way.
Simply envoking the title of the office doesn't mean the act falls within the lawful powers of the office.
Think of it like the police. An off duty cop is not acting under official powers. Even on duty, a cop is limited in what and how they are supposed to do their job and crossing outside those bounds technically makes them personally liable. I wouldn't begin to say we are any good at actually holding police accountable when they act outside their official duties, but that's a whole separate can of worms.
Meaning, if the president shoots a random bystander on the street they can be charged with murder. If the president orders a military strike as part of an official operation and done through proper channels, they can't be charged if it later turns out the intel was bad or the strike went wrong in some way.