555 Timer in monostable mode with reset on the same momentary button
Rather than trying to delay operation of the reset function, you're better off trying to delay the trigger function. It's not a bouncing issue, simply that as soon as the RESET input is activated, the relay turns off and the TRIG line is activated, turning the relay on and starting the cycle again. You've effectively made an oscillator.

The key difference between that circuit you found and yours is the capacitor on the RESET line. Pressing the button to turn the circuit off discharges that capacitor through the button, and it prevents the TRIG input from having any effect until it's charged back up again, and that doesn't start until the reset has taken place and the relay has switched.

The circuit will still oscillate, but much more slowly and controlled by the RC values used on the RESET input. One way to prevent this altogether would be to use a more complex circuit with the button used to clock a flip-flop, but I would think an RC delay of two or three seconds would probably be good enough for your application.