Greetings!
I have a Fiat Stilo Multiwagon JTD 2003 left hand drive.
I've found a very good picture of where the switch is in a Spanish forum.
Procedure:
Remove the driver side front wheel (left).
Look up behind the shock absorber and under the battery. A torch light might be useful.
You will see the cable (white and brown wires).
In the picture it is the one in the green square.
Open the clip of the plug with the help of a screw driver. Put the cable out of the way.
Then remove the switch with a 19mm ring wrench rotating it counter clockwise.
It must rotate more or less easily.
Insert the new one from ebay (3x cheaper than in Fiat).
Assemble all back and the reverse light will work again.
The old switch didn't have any damage so it must be an internal thing. Odometer is 177000km.
I've used an empty volley ball to lay down my head while I was looking up.
The position is quite tiring to the neck.
My regards