What kind of insurance pays to fix damages that you cause but does not cover your own car?

If you were driving someone else's car with their knowledge and consent, and were named as an additional insured or otherwise covered under the policy, that liability coverage is likely to pay. If you neither had permission nor were an additional insured (by being named on the policy or because of a relationship to the named insured), you probably would not be covered.

As a practical matter, if the damages were relatively minor, the insurer may make a business decision to pay the claim. Otherwise, it would issue a "reservation of rights" to the named insured stating that it was not going to cover the claim.