I've finally found the solution, so here is the explanation in case someone else gets the same issue:
I noticed that I have 2 VGA ports on my computer: the cable needs to be plugged in to the graphic card VGA port, not to the motherboard: otherwise, there is no signal. My computer turned down immediately after pressing On/Off button because after replacing the CMOS battery, the BIOS was reseted and it asked what to do about it: reset default and restart or reset default and continue boot. As the BIOS didn't even launch, the computer directly shutted down when button was pressed. As soon as I had access to a signal on my monitor, it was easy peasy to correct. Finally, the initial problem came from a windows update (I didn't remember it updated last time I used it) that went wrong. A little restoration and voilà, brand new!
Good luck to anyone having the same problem.