The word that thrilled the president and his legal team was "subject."
It seems that Mueller and one of Trump's lawyers had been negotiating the terms under which the president would submit to an informal interrogation by Mueller and his team of prosecutors and FBI agents. Mueller's request for such an interview was not unusual.
Investigators are usually looking to trap an unwary potential defendant into lying to them — a crime in that environment, even though the potential defendant is not under oath — or unwittingly admitting to them an allegation for which they need proof. The potential defendant often believes — foolishly, as history has shown — he can actually talk the investigators out of indicting him.
In preparation for this type of interview, the government often tells the lawyer for the person being interviewed whether that person is a witness, a subject or a target. A witness is a person whose knowledge and memory the government wishes to examine. A subject is a person whose behavior is under criminal investigation. A target is a person whom the government plans to indict.