Um...no it doesn't
Hate to break it to you, but what you just described sounds like a problem with your ABS system. That's NOT how it should work at all. You might want to try to fix that.
This is not a Mustang thing. That applies to ALL vehicles. The ABS light comes on during startup to tell you if the bulb is functional and then shuts off until a code is stored in the system. When a code is stored, the ABS light comes on and the system is disabled. During ABS use, the light DOES NOT come on at all. You will never see the YELLOW ABS light at all UNLESS something is wrong.
The traction control system (which is part of the ABS system BTW) does display a spinning wheel when the system detects wheel-spin and activates. That's also not a Mustang thing...all cars equipped with TCS do it to tell you if the system activates.
If you are braking hard, activating the ABS and the light comes on, that means your system is finding a fault somewhere during use and deactivating itself. Also, the car shouldn't feel violent during ABS use. Other than the brake pedal vibrating, the car should track and drive fine with no vibration in the car itself.
I'm not trying to be a jerk here...but your ideas about how ABS systems work is incorrect
PS. If the red BRAKE light comes on during hard braking, it means you are low on brake fluid
