You’re certainly on to something with how the question is phrased. We won’t know the answer until they rule.
However, alleged in this context means alleged by the prosecutor (the indictment). How it generally works is the prosecutor will file their indictment that will state how the prosecutor thinks (alleges) the defendant broke the law.
Then, the Defendant can immediately file a motion to dismiss the case, and basically say, based on those facts in the indictment, I’m immune from prosecution, it’s past the statute of limitations, or whatever other reason. The key is the Judge can only look to the alleged facts in the indictment to determine what happened. The defense cannot offer any new facts. very generally speaking.
In this case, the indictment is thorough and the facts are well laid out to allege the defendant committed the crimes.
Later, once more facts are proven and were not just relying on the alleged facts in the indictment, say after the trial but before it goes to the jury, the defense can ask for a motion for directed verdict and use those new facts (actual evidence) to say the case should be dropped, etc.
That’s how I take the word alleged to mean in the SC question. I don’t think it’s a big deal and it’s normal to use that word at this stage of a case.
The rest of your post, about what constitutes an official act is a good question. Some we know for sure are official acts since they are spelled out in the constitution (treaties, sign bills, commander in chief during war, etc), some we think we know (?) and some we know are definitely not (?). There is an intuitive sense of what it is, but I don’t know so I put a ? and I think it’s what the SC wants to find out.
There is some form of immunity for the President from criminal prosecution - everyone agrees to that (at least Trump and DOJ do). But disagree to what extent, how far reaching it is.
Finally, I’m confused by limiting only to official acts because I’m thinking the indictment tried very hard to lay out everything as not being an official act - it was personal/election related type acts/done in concert with civilians, etc.