New try to burn the bootloader. This time, the led13 was flickering but the process ended with a new error:
avrdude: stk500v2_recv_mk2: error in USB receive
avrdude: verification error, first mismatch at byte 0x10f0
0xff != 0x00
avrdude: verification error; content mismatch
I ignore the real signification of this message but I think I read somewhere in the forum not to pay attention to it.
From Studio 4 through the avrisp mkII programmer, I made a verification of the bootloader (ATmegaBOOT_168_atmega328.hex). There was no error at all. So I suppose the error after flashing the bootloader was not real but had something to do with the trick of the signature.