Viewing previous processes/application activity on Mac
Not really - and the more you try to prove with technical details, you might be sabotaging yourself. It’s easier for you to tamper with the process list than it is to cheat on an online test, so I wouldn’t go down that path at all.
I would just ask the accuser or referee how you can demonstrate mastery of the material and remedy the situation. It’s awful when your character is accused and it’s awful when you get caught cheating, so either way, you are in a bad situation and grasping for a technicality isn’t going to help you. This is a social problem, not one that can be dismissed without two conditions:

Referee has to be agreeable to a technical summary and adjudication.
You can in no way be involved in the preparation of any exculpatory data. A forensic investigator has to handle the preservation, preparation, analysis and evaluation of any technical evidence.