Battery capacity is defined by SERIES count. A 12S battery, fully charged, is 12 times 4.2V, which is fully charged voltage for almost all lithium cells. 12S gives 50.4. This is what is referred to as a "48V" battery. 13S is 54.6V and is called "52V". 14S is 58.8V and is called various things.
Notice that the numbers are NOT an average, nor any reasonable value to define a battery. They are simply easy approximations that honestly do great harm and are stupid. Many ebike fires are caused by this idiotic convention, and the end users lack of any knowledge whatsoever of what it means.
YOU MUST KNOW YOUR SERIES COUNT.
Overcharging a lithium battery is extremely dangerous.
BMS and Controller have LIMITS as to maximum amperage, but not a CUTOFF. BMS and controller have CUTOFFs for Low Voltage. SOMETIMES these are programmable, SOMETIMES they are set by a changeable menu item. This is for the controller, not so much for BMS, meaning the programmability.
If you have a 30a limit, then the controller or BMS will not put out more than that, at least continuously, but they will not shut down.
Battery BMS cutoff is reset by turning battery off and on, which also resets controller. Controller is reset by removing power. Removing battery resets the controller. Leaving battery switched on while doing a remove and replace resets the controller but not the battery. Note that there is some risk involved installing a live battery.
If you are cutting out at 50V, then your controller is set for a MUCH higher voltage than your battery is capable of.
Battery BMS LVC are almost always set up for the battery they are installed in. Controllers are not closely connected to the battery voltage when built or installed. They are much more likely to be incorrect for the existing setup.
Oh, and you have established pretty much beyond doubt that, in fact, it IS the "BBSHD programming". Did you buy this from Luna, that is the kind of mistake they make pretty often.8