Generally, if you ding a rented aircraft and it is an 'all risks' policy then you can usually reckon the limit of financial damage to yourself will be the 'excess' on the policy, eg 500 or 1000 pds, whatever has been agreed.
However, if the damage to the aircraft has been caused by some lapse on your part, eg ran out of fuel, then while the OWNER of the aircraft may be indemnified against such risk (and get his repairs paid for by the insurance company) YOU as pilot may well be faced with a zealous insurance company wanting its money back. It is called subrogation and is likely to follow any successful prosecution by the CAA for said lapse. This is on the basis that you cannot insure yourself against the consequence of an illegal act on your part .. and if you are successfully prosecuted you must have committed an illegal act.
There is a young lady who found this out to her cost a few years back with a bent Baron in the Bournemouth area. She had to reimburse the insurance company some 85K I seem to recall.
So .. be careful where you run out of fuel or commit one of a host of other aviation transgressions or it might not just be a bent prop for which you end up paying. When third party liability costs are involved the figures become absolutely frightening.
I can not see any difference whether you are a student or instructor if you are in command. I am one of the latter and confess I just shut my mind to this - it is just too horrible to contemplate for long. And I would love to be contradicted by a legal-type out there. Any takers ?
Caveat aviator.
Isn't aviation a wind-up ?