What actually kicks in a VFD?
Most VFDs are like solid state relays in terms of what turns the motor on and off. The motor is turned on and off with solid state switching devices, IGBT's in most VFDs, but other devices are used as well.

VFDs are different from solid state relays in that the switches are constantly switching to create the AC output waveform. That means all switches are off when the motor is off and all switches are sequencing when the motor is on.

Also, the motor is not just turned on at the operating speed, it is turned on at a very low speed and then increased over time to full speed. Similarly, for a normal shutdown, the motor is slowed down and then stopped.

It he additional safety of a contactor is required to disconnect the motor, the contactor is closed before the solid switches are started and opened after they stop. A contactor could be used to power the VFD on and off, but that is not recommended to be done more than once a day or once a work shift.

When a VFD is shut down to protect itself, that maybe done by shutting off all solid state switches instantaneously or by decelerating the motor first. The stop mode would depend on the type of fault.

The different ways of controlling the VFD may be determined by manufacturer design or customer preference.