If the FO can't call stop what is he doing there. Let's make it a single crew operation and make it easier for everyone. If your company training is so poor that a professional pilot can't make a Stop/Decision maybe there's a need for an overhaul in your trg dept.
Your comments do not appear to be those of a pilot in command, or to have been made with the benefit of PIC experience.
The operator for whom I presently work requires anyone in the cockpit (Captain, First Officer, or FE) to call out the malfunction loud and clear. The captain makes the decision to reject the takeoff. There is no "stop" call. (In fact, I've never heard it used in any company for whom I've worked, or training I've undergone).
If the captain is performing the takeoff and elects to reject (in accordance with the prebriefed criteria for the takeoff), then the captain rejects. The flight engineer backs him up on the reverse and speed brake lever. The first officer notes the airspeed (for brake energy calculations later) and notifies the tower that the takeoff is being rejected.
If the first officer is performing the takeoff and the captain elects to reject, the captain takes the controls while announcing the reject. The flight engineer backs him up on the reverse and speed brake lever. The first officer notes the airspeed (for brake energy calculations later)and notifies the tower that the takeoff is being rejected.
If the captain is performing the takeoff and the captain elects not to reject, the captain continues flying the airplane. If an engine failure has occurred, the captain proceeds to fly the engine-out profile. As soon as practical, the captain will usually turn the airplane over to the first officer to fly, in order that the captain can work with the flight engineer in addressing the problem.
If the first officer is performing the takeoff and the captain elects not to reject, the first officer continues flying the airplane. If an engine failure has occurred, the first officer proceeds to fly the engine-out profile, as briefed.
When an abnormality occurrs on the runway, we either continue or reject the takeoff. There's no "reject" call, other than the captain announcing that he's rejecting the takeoff. Any malfunctions are called out loud and clear, but in general terms. When the engine fails close to V1, we don't care if it's #1 or #4; the gist of it will be apparent through the necessary footwork, so the call is simply "Engine Failure."
Generally the brief is conducted along the lines of "We will consider rejecting the takeoff for any problem under 80 knots. Between 80 knots and V1 we will reject only for engine fire, failure, or loss of directional control, an outward opening door, or anything which brings the ability of the airplane to fly into question."
Given that rejecting a takeoff, especially when heavy, can be one of the most dangerous acts one can attempt during the flight, the responsibility to make that call and take that action for us rests with the same person who assumes the ultimate responsibility for the airplane and the safe outcome of the flight; the PIC. The first officer does not instruct the captain to reject and the first officer does not make the call. The first officer calls out the malfunction, but the captain decides what to do about it. Much of the time, especially approaching V1, the decision will be to let the first officer fly it off, and then handle it in the air.