The power button for the TBT4-UDZ is simply to turn the dock on or off. This button does not have any influence on the computer itself (power button will not power on/off the attached host system).
When the dock is powered off all devices attached to the dock will also lose power and the dock will no longer be recognizable by the system or offer charging (until it is turned back on).
Please note that the dock will automatically power on when attached to AC power and this is indicated by a white (or green) light on the front. A white light means the dock is getting power but that it is not connected to a host system. A green light means the system is getting power and has successfully connected to a host system.