Y's insurance. He owns the vehicle and gave permission to X to use it,
so he's the responsible party when it comes to insurance coverage for
what happens to/with the vehicle.
We had this come up in our family, when a relative borrowed another
family member's car and promptly had a fender bender. He first claimed
his insurance would take care of it, then offered to fix the car
himself. After he failed to do so, the vehicle owner phoned her agent to
ask whether to file the claim with the relative's company or her own.
The agent explained that the relative's insurance company was not on the
hook, and that she would have to file a claim against her own policy,
for the reasons outlined above. Another family member felt that was a
horrible way to treat someone who'd tried to be kind by loaning her car,
so he stepped up and paid for the repairs out of his pocket.
You can imagine how tense the family gatherings were after _that_.
If I recall right, this is one of the reasons why insurance for everyone
can go up like crazy if another person at the same address has a dui, or
accident, or other moving violations that effect insurance rate. The
idea is that you are more likely to loan the vehicle to a roommate or
family member that you live with and they'll pretty much have to insure
anything that person does to the car as well :).
Andy