On the older types like B707,727,737,747,757,767, A300B4,A300-600, MD80,82,87,88, MD11, F70,100, You are right.
The newer Aircraft like A320,330,340, B737NG,744,777 you have a acoustic and optic alert, EICAM / EICAS message that recognizes the diffrences in N1/EGT/FF/EPR and alerts the crew to shorten the recognition time and the reaction time. Before V1 remains the same = ABORT the T/O. After V1 if you abort the T/O then the chance is high that you end up in the mud and the opposite approach lights are severly dameged as well as the hull. After V1 it is safer to go and then return to the field. That is why you also require a T/O alternate if the field is below CAT I. If you decide to abort the T/O after V1 you will have a lot of to explain..... In a SIM-check that is usualay a failure of a P/C as the risk management is badly handeld, even if you manage to stop before the end of the RWY. Any futher question?.
NG