Is there a standard definition of the "stall" phenomena?
The scientific definition of a stall is as you state in the first line, however in the cockpit that definition isn't actually that useful. Practically there's 2 parts of stalls a pilot needs to be aware of because they are things a pilot experiences, and impact the flying characteristics of the airplane:


Stall buffet: this isn't a stall, but it happens when you're very close to one. Recognizing a buffet may lead to a pilot decreasing AoA before entering a fully developed stall
Flow separation: when the flow separates from the top of the wings it may do so on one wing before the other, causing a wing drop, aileron input may work opposite of intent, and you of course have a high rate of descent. This can feel mild or violent depending on the airplane's characteristics and the speed at which the stall happens


Whether the definition of a stall is when the airflow separates or a change of the center of lift is academic, what a pilots needs to know is how to recognize the condition and recover.