His problem is slow or no cranking when hot.
Problems with these wires on fiat very common.It may be that, but the car had the same problem when it had 60k kilometers(when I bought it, maybe it had the problem even when it had less km).. now it does it more often tho, and sometimes doesn't start which haven't happened before..
On the starter there is a connection for a wire that triggers the starter to work.
Put an extra wire on there and when it doesn't want to work touch your new wire to battery +ve . If that causes it to work right then you have problem with trigger circuit.
That does not test the circuit that causes starter to operate, fit an extra trigger wire , try it and report back.As I said earlier, I used a jump starter and I connected the cable directly to the starter, not through the battery, i got under the car(where the starter is). But when I took off the starter to test it, it spun so the starter is not a problem(i tried with two, I had them checked to a electrician)
You have faulty cranck sensor.