According to Boeing either crew member should call out the malfunction, who ever sees it first. Regardless if it is the captain or not.
Only then will the one that holds the throttles (the captain in the case of Boeing) call stop. It would be pretty stupid to have the first officer call stop, as long as he is not holding the throttles.
The benefit of having this responsibility placed on the same person on every take off, is that there will be no confusion as to who should do the abort and the captain will always be in the state of mind to perform the about. The maneuver itself, while uncomplicated, is (should be) a reaction based on memorized reflexes, hence his muscles are instinctively ready to perform those reflexes on every takeoff. If the responsibility changed with every takeoff, the brain would have to process who is to perform the reject before the muscles take action causing an unnecessary delay.
But as said either can call out the malfunction, and the rules are clear cut when a rejected takeoff should be performed or not. The first officer calling "engine failure" before V1, is effectively the same as calling stop. The actual "stop" callout from the captain is a confirmation that the actions are being taken.
Only ambiguity in the rules are the "unable or unsafe to fly". Which in my opinion is a ridiculous wording. Unless you had a collision on the runway where a wing is torn off, it's pretty much impossible to say if the aircraft will fly or not until you reach VR in which case you would be out of luck with regards to stopping anyway.