So ‘no immunity for unofficial acts’ is synonymous with ‘no immunity from illegal acts’?

Not synonymous, no. For example, Congressmen are immune for anything they say on the Congress floor - even though it's illegal to say slanderous or libelous things. Some of their "official acts" are immune, even though they are ostensibly illegal. I say ostensibly, because if pressed the courts would certainly rule that those laws would be unconstitutional, as applied, to Congressional floor speech.

But none of the parade of horribles that folks have been discussing are official acts, because they are outside the powers of the Presidency. The President doesn't within his powers the ability to assassinate a U.S. citizen on U.S. soil. You can make up a fanciful scenario where the President successfully lies to a court about some crazy set of circumstances that would ostensibly bring it within his powers - but if you do that, you're just constructing a scenario where the President would have the charges dismissed even if he weren't immune. Which would mean the President has always had the power to assassinate political opponents with impunity, because the courts will just believe his lies....

....which is just silly. I mean, the President tried to argue that all of his acts were official in this case, and this Supreme Court rejected that argument. They refused to accept that everything the POTUS does is official, and instead sent it back down to the lower court to make a determination.

Legislative acts have been immune from prosecution for centuries, and that hasn't stopped Congressbeings from being tried and convicted for crimes like bribery and conspiracy regarding their legislative acts. Nor has it resulted in an overwhelming wave of Congressional lawbreaking. Regardless of the merits of the ruling, the effect of the ruling is not to authorize Seal Team 6 assassinations, as intimated by the dissent.