there is a starter relay recall on these cars you need to check with dealer to see it is has been done. It adds a relay mounted on core support to take the load off the ign switch. it is possible the the recall has been done and ign switch was ok at that time but now the ign switch has gone bad. The back up light has nothing to do with the starter circuit I would wonder it you have a loose connection on either battery cable at block or starter or a problem with the control wire at the starter
attached is a copy of the recall notice 2006-06-02_131250_relayrecall.zip
I hope I have been helpful. If more info is needed just ask.
Please select
Accept when your question has been answered. And
Bonuses and Feedback are nice too.
Thanks Tim