Emergency brake (hand brake) light problem

Hello
I have noticed something in my Mazda 3 '04 that I do not know if it is normal or not since I bought it recently. The emergency brake warning light stays off when the key is in the ON position despite the fact that the brake is engaged. However, if the key is not in the ON position, when all the lights go on before starting the car, the warning light is shown.
Anyone any idea of what could be the problem?

Thanks for your help.