The point of presidential immunity is that it is based on the action taken, not the motive behind the action. Motive would be part of the mens rea criminal inquiry that would kick in only after it was determined that immunity does not apply. But if immunity applies based on the nature of the action, that's the end of the inquiry. You never get to intent/motive.
One line you quoted from the syllabus sums it up unambiguously:
"In dividing official from unofficial conduct, courts may not inquire into the President's motives."
Making motive part of the test for immunity would defeat the entire purpose of having immunity in the first place.
Again, thank you.