As it appears in Wordreference.com.

another option!
9.4 In the event of the occurrence of a Termination Event or Lessee Damage, if Lessor elects not to restore the Building or improvements it will give Lessee ...
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SEC Info - Pharmaceutical Product Development Inc - 10-Q - For 9 ...- [ Traduzca esta página ]9.3.2 “Lessee Damage” means any damage or destruction caused by the ... 9.4 In the event of the occurrence of a Termination Event or Lessee Damage, ...
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AGREEMENT OF LEASE- [ Traduzca esta página ]Upon written notice to Lessee, Lessor may terminate this lease due to Lessee's damage to said leased demised premises. Lessee further agrees that the Damage ...
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Today’s Date: __ 2006 Space Rental Lease ___This Lease agreement ...- [ Traduzca esta página ]Charges will come out of the Lessee's damage deposit. ... If conditions of the lease are met, Lessor will mail the Lessee's damage deposit check to the ...
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Note added at 26 mins (2007-06-05 09:22:02 GMT)
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LOCKYOUR CAR AND PARK IT INTHE AS IGNED AREA. PLEASE DO NOT LEAVE CAR IN DRIVEWAY. ..... and the cost for such repairs shall be paid by the lessee. Damage ...
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SEC Info - Asia Satellite Telecommunications Holdings Ltd - 20-F ...- [ Traduzca esta página ]Car parking spaces (24) Not to use that part of the said premises built or ..... as the case may be by the Lessee. Damage to (42) In the event that, ...
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In insurance field, the FULL RISK insurance is about policies which cover not only the damages caused to third parties, but also the damages suffered by ourselves (own damages)....which normally would not be covered on a regular policy.
When I worked for Royal&Sunalliance Insurance Co. we in Spain used the term "todo riesgo" (full risk) to refer to these policies of covering own damage besides other coverages of the policies (water damage, fire, third party liability, robbery/burglary....etc.).
In this context, there seems to be not insurance involved directly, but indirectly.

That is, the rental agreement is specifyin that any damage due to own liability its entirely under responsibility of whoever is the "arrendatario". or else, that the "arrendatario" is responsible to hold liability, directly or through any type of insurance, against own suffered damages

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Note added at 27 minutos (2007-06-05 09:23:24 GMT)
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own damage liability can be applied either to insurance field as well as in legal documents which do not include insurance.
If in insurance, this particular cover can be applied either to motor insurance as well as other type (business insurance, home insurance, and others)

Why not work in the correct English term for damge to your own car/personal injury and that of third parties. It would mean re-structuring the sentence but it would be much more eloquent. Personal liability also springs to mind.

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Note added at 26 mins (2007-06-05 09:21:34 GMT)
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PS the 'not' in: Please note this is not a car rental agreement is a typo right?

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Note added at 27 mins (2007-06-05 09:22:33 GMT)
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If the not is correct, personal liability would still fit I feel.

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Note added at 38 mins (2007-06-05 09:33:55 GMT)
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As this is insurance for leasing generators - I'd go for: ...Personal injury, Personal liability and Public liability cover will be the responsibility of the lessee... or something along those lines.

Another suggestion

I think you could stick to simply saying something like this:

.........damages caused and damages to third parties.