Do you realise that the DC controller circuit in the link provides instant reversal of motor drive direction with no dead zone at which the motor is 'off' ? That is bad for the motor.
The proposed modification to the circuit makes no allowance for mains isolation and is potentially LETHAL.
Also, it is unsuitable for an AC motor. If your motor has only two wires then it is not reversible.
I suggest you research 'solid state relays', which can provide both switching and mains isolation. I would also advise using a control circuit which provides for forward/off/reverse control of the motor (if indeed it is reversible).