Welcome to ECN, mate.
To add to what John said above, the way that the 3 wire Start/Stop works is this:
Effectively, what happens is, you run a 3 wire control cable from the supplying DOL starter box to a remote Start/Stop station, this could be on the other side of a machine, but it would also need to be within sight of both the original starter box and the operating parts of the machine in question.
What happens from there, is the remote Start circuit is wired in parallel with the original Start circuit, this also encompasses the "hold-in" contact on the motor contactor, this ensures that power is continuously supplied to the coil of the contactor.
Start contacts are always wired NO (Normally Open), this to ensure, with a motor or equipment that vibrates a lot, if the Start wire breaks, the equipment cannot be started until that equipment has been repaired and re-commissioned.
On the Stop side, the switches are wired in Series, this comes from the control circuit side of the thermal overload, so you have a continuous loop of switches from the start of the Stop Circuit back to where it ends at the Stop circuit.
Stop contacts are ALWAYS wired NC (Normally Closed), this is to ensure that if a wire breaks in the Stop circuit, the machine will just stop, as you've lost power to the control circuit.
E-Stops, Guard limit switches and light curtains all use this method, the effective result is the same.
The "beauty" of using a contactor control system with motors, is the fact that on loss of power, the contactor will drop out, as in if the "hold-in contact" can't be maintained through lack of voltage, it will need to be manually reset.
This is what is known as a "No-Volt Release".
It also stops motors from suddenly re-starting on the restoration of the supply, in process control and some other things, this is VERY important.
Hope this helps, mate.