Ok, I bought a new power supply which is regulated 600 mA 12V tip-positive. I checked the "no load" at 12V on the power supply and it gives me exactly 12V so, as Wilba suggested, it is supposed to be regulated.
I set the JBL shunt on 12V position, then I put a shunt on JR4, no shunts on JPB and the JP shunt in 6581 position. I checked the voltage in orange points and it gave me 10,5V.
Once connected the two boards the LCD came to life as I saw backlight and the "squares" displayed on its upper row, but after 5 minutes the backlight started to blink and in the end it went totally off; I can still see the squares row, so the LCD is working.
As far as I can check I cannot understand where the problem is. Is the LCD backlight definitely dead?
As I have two 6581 and a 12V regulated power supply I have to set both the two shunts in JPB, right?