The best example is first responders. They have immunity doing their duty. They cannot hesitate to perform their duty - such as giving life saving services - if they fear they are unsuccessful and are sued / thrown in prison. If they break the law though on duty it was never their duty to break the law and are therefore not immune. Take CPR. They might perform CPR and injure the person they are working over, or they might not save them. The family of that person cannot sue them, nor can a court convict them if they accidentally make things worse.
Same thing with the president. The president can’t break the law and say, “whoops, just doing my official duty”. It doesn’t work like that.