I'll bite on the definition of authorized instructor. In the OP's scenario, who signed the student's logbook? I bet the MEI up front did. Now using some common sense, I would guess no one has a flight that was endorsed by two CFIs and I doubt any of us who have a CFI would a.) see the need to have two instructors sign for one flight or b.) actually be willing to sign for a flight after someone else put their name and number there.

My conclusion is that if you wouldn't be willing to sign for the student, or don't sign the student's logbook stating that you did give them instruction, you shouldn't log it in your own logbook.