A power supply unit (or PSU) converts mains AC to low-voltage regulated DC power for the internal components of a computer.
Turning on a power supply is done by a signal from the motherboard through its power plug. Indeed, testing a PSU when it is not connected to anything can be done by shorting two appropriate pins. In the packaging of my Seasonic Focus PX-850, there is a “power supply tester” that can be fitted to the 24-pin motherboard connector, shorting the two appropriate pins.
(I bought the above Seasonic PSU in june 2020 to build a little workstation.)
It seems a rail is a one or more groups of cables (and thus connectors) that share the same circuitry and current sensor. Multiple rails can be present in a PSU to split the available load (sometimes called power distribution), so that a single rail failure offers a limited risk.
Conversely, a PSU with a single rail is simpler to use as there is no need to balance the load.
(My Seasonic has a single +12V rail, delivering 70A (i.e. 840W), in addition of other minor rails).
I was wondering how a BMC could be running on a motherboard when the PSU was switched off. The ATX specification includes a +5V standby voltage, present as the pin 9 (purple) of the 24-pin motherboard connector.