How does Battery Management System work (at the hardware-interface level) ? And how is idle self-discharging of batteries handled?
Battery Management System is a very broad term that covers battery charging /discharging/monitoring / power IC /responding to critical events like shutting down on low battery level / User Interface actions (increasing CPU speed when user touches screen - touch boost) and many other things.

Your first quote is only instance of management system where the battery voltage has fallen to a very low level, and a protection circuit kicks in and prevents further charging. This protection mechanism is mandatory


To prevent the battery from over-discharging, a control circuit cuts off the current path at about 2.20V/cell.


Booting up from a fully drained battery, is not permitted, since it would only further drain the battery. Booting  consumes a lot of power and doing so on an already drained battery can lead to irreversible damage to the battery. And what's the point of booting up, if the device can't be used thereafter?

There are ways to overcome these problems but it is best left to  qualified technicians to handle it. There are many questions on this site dealing with such situations.
As you would guess from the first  paragraph, this being a complex system, it is dealt at a low level, meaning at a kernel level with proprietary hardware interfaces. Related Where are the battery capacity files located?