I don’t think that’s an accurate description of the decision. I think that it stated that when the President exercises core constitutional power (e.g. the pardon power, or the veto power) then the exercise itself cannot be illegal. I’m not sure if the decision of the Court left open the possibility that the conduct around the exercise of such power can be illegal. If so, then this could be a distinction without much difference: for example, issuing a pardon may not itself ever be criminal, but taking a bribe to issue a pardon is separate from issuing the pardon itself. To some extent, I think that some of this does flow from the structure of the Constitution itself, but I’m not convinced that phrasing it in terms of immunity is particularly helpful.

Then there’s a rebuttable presumption of immunity for more conduct. I don’t see that this flows from the Constitution, but perhaps it flows from judicial decisions over the past two centuries? ‘When the President does something official, he probably is immune, but maybe he’s not, and he could still be prosecuted from crimes he commits around the immune act’ doesn’t seem terribly meaningful.

It sounds a bit to me like saying that a citizen is immune from prosecution for his vote, but not for selling it or whatever. But I’m not a lawyer, and I could be wrong.