I think that's a bit unclear. They have this standard where things that are part of a presidents "core constitutional powers" enjoy absolute immunity and anything else enjoys "at least presumptive immunity". I think this hints that some members of the court wanted to go further, but anyway I'm not quite sure what "presumptive immunity" means in the context of the law, but I think this quote gives the most clarity for me.
> unless the Government can show that applying a criminal prohibition to that act would pose no “dangers of intrusion on the authority and functions of the Executive Branch.” Fitzgerald, 457 U. S., at 754. Pp. 12–15.
This seems to be the real boundary. To show something doesn't have immunity you either have to:
1. Show it was an unofficial act
2. Show it wasn't part of the core presidential powers and that prosecuting it wouldn't have any danger of intruding intruding on "the authority and functions of the Executive Branch."
I think that last standard is related to presidential immunity from subpoena which courts have also recently construed quite broadly