Hi iVan_sbk
Don't worry about how much current the heated sreen is drawing, it will draw what is needed. A heated screen of this size draws between 8 and 12 amps. Your problem is with the connector, the contacts look pretty small which means that they have to have optimal contact pressure to make sure that they do not start heating up due to the contact resistance with this amount of current. This is obviously what has happened ( FIAT electrics?!?! ). Once electrical contacts like this start to burn you just have to change them. Once the surface of the contact starts to burn, its resistance goes up and makes for even more electrical heating of the contacts ( I²r ). A vicious circle which could lead to an electical fire!! Since the heated rear screen circuit is probably protected with a 30 amp fuse, there will be enough current to do some damage before the fuse blows, if at all.
The answer to your problem is to cut off these two connector ends ( noting which wires go to which ) and replace with standard quarter inch flat blade connectors. Remember that there's quite a lot of current flowing so make a good job of it!
Chris C