Whether you own, lease, or have a loan on a vehicle, your insurance policy is going to cover the vehicle the same based on the coverages you choose.
Many banks will require you to carry full coverage on a leased or financed vehicle, but when you own a vehicle, you have the option of carrying comprehensive, collision, or just insuring the vehicle for liability. When a vehicle is financed or leased we will add the lienholder/lessor as an additional interest so that if you experience a total loss, we can pay off the loan or lease with the company.
The only additional coverage I would suggest you consider when you have a loan or leased vehicle is GAP coverage. GAP coverage pays the difference between your loan amount and the value of your vehicle at the time of a total loss.