Weird: Pro Mini sometimes won't start after uploading with an USBasp programmer

Azzy, seems that you're definitely right.

Below what is reported in the main page of the Minicore libs on GitHub:

"Note that you need to select Bootloader: "No" and actually hit Burn bootloader in order to set the correct fuse bits.
Now you're no longer able to upload using serial, until you select Bootloader: "Yes" and burn the bootloader again."

Shame on me.... :-[

Seems that all I have to do is:

Step 1: press "Burn Bootloader" to set the fuse bits.
Step 2: upload the sketch as before.

Now I cannot access to my little "laboratory" to test this solution. I have to wait until tomorrow, I will report my results here.
I really hope that this is the reason of the problem. I cross my fingers!