The controller uses a overall battery voltage for LVC. The BMS uses individual cell voltages often as low as 2.5v.
When the cells get down as low as 3.1v, they can't keep up with the bike's demand for current, and they go crashing down. The BMS will switch off when the first cell reaches LVC.
The more out of balance your cells are, the higher the voltage you would see for the whole battery pack when it cuts off. You could cut off at any voltage if you have any dodgy or unbalanced cells.
I've tested many batteries with a proper ebike battery tester. Good healthy batteries usually cut off at around 29v when discharging at 5A, which is what I use for capacity testing. If you discharge at 10A, the cut-off voltage would probably be higher. The return testing voltage would definitely be higher.
The battery voltage is very dynamic. It can change by several volts, depending on the amount of power you're taking from it and its state of charge, so none of these numbers are fixed as far as the rider is concerned.