While backing up, you run into the garage door you just installed. Should you submit the claim to your auto or homeowners insurance carrier?

Because you own the garage, damage to the door will not be covered by your personal auto policy. The ISO form reads:

EXCLUSIONS
A. We do not provide Liability Coverage for any “insured”:
2. for “property damage” to property owned or being transported by that “insured.”

However, your homeowners policy will cover this damage, although subject to the policy deductible.