Welcome to Just Answer. I am here to help you resolve your tax and finance concerns. Please feel free to ask anytime you need extra help.
Since there is no tax advantage to either choice, I will give you my opinion from a financial standpoint. I would have the policy issued under his name. He, at 19, is statistically more prone to an accident than you are. If such an accident occurs he will be liable not you. At this stage in life you have much more to lose if there is an award beyond the insurance coverages. (In a world where a MacDonald's customer is awarded $5,000,000 because he spilled a hot cup of coffee on himself you cannot always count on your insurance to cover everything). He is also more able to file a bankruptcy to eliminate claims against him and stands to lose less by doing so if ever needed. I would have excellent coverages - similar to those you have on any other policies you may have and meeting, as a minimum, the coverage required under the lease. Doing this will cost more but will provide your family the safety you will not have the by putting the policy in your name while he is the primary driver of this vehicle.
I have personally been with State Farm since I was 16 (62 now). My father wanted exactly what I told you. At the time I did not see the wisdom of his choice (he did hand me the bill) but after all these years I see how it was necessary to protect the family home, his pension, and everything else.
As to my insurer, two months ago I was involved in a 1 car accident which I labeled "Adventures in Offroading with Charlie" during an ice storm. The insurance company stepped up rented me a car (I paid 20% of the rental but State Farm also got me a discounted rate so my total bill for 30 days was $187.66). They completely repaired my car and guaranty the repairs as long as I own the vehicle. I do recommend them. The minimum state requirement companies that advertise on TV, while appealing, do not satisfy my desire to protect my home and family, get excellent service (always have), and all that for an annual premium of under $900. Can't beat that. Your son's policy will cost more because of his age and the statistical risk associated with that but what is the value of your future?