Emergency shut-down can have different categories, or levels, to determine whether the shutdown is instantenous or controlled, controlled meaning allowing the machine time to come to a complete stop then disengaging by electromechinal means.
Just a couple of years ago standards, in the US, stated that operation of ESTOP could not depend on electronic logic, software or hardware, or commands over a network or communication link.
Some of that has changed with the use of Safety PLCs etc, where it may be allowed to use the plc to control the machines stop...it depends.
To answer the original question to the best of my ability; ESTOP is a procedure to bring a machine (system) to a halted condition, in many cases this will be a hardwired solution BUT it may be possible (now) to have a PLC incorporated into that procedure.