On a Boeing, one aborts an engine start, and one rejects a takeoff.
At our company, the Captain makes the decision, and executes, a rejected takeoff. If the first officer is performing the takeoff, he stands the power levers up, moves them toward the takeoff power setting, and then calls for reduced thrust or maximum thrust.
At that point he removes his hand from the thrust levers because he's done with them. The captain guards the thrust levers until V1, when he removes his hand, too.
If an incident occurs in which the captain elects to reject the takeoff, he will anounce that he is rejecting the takeoff, and perform the RTO. In the absence of the captain taking the aircraft for a reject, the FO will continue to fly the airplane until instructed otherwise.
We have clear instruction from the company on the matter, as well as specific counsel that there is nothing on the forward annunciator panel worthy of rejecting a takeoff.
We brief the situations in which a rejected takeoff will occur as part of the departure briefing.