If you trade in a car or put cash down on a lease to lower your payments, you're just throwing money away. It artificially lowers the monthly lease payment by paying a chunk of money up front. Why pay the monthly payments early? It makes no sense.

For example, say you trade in a car and the dealer gives you $5,000 for it, and your monthly payments without the trade in are $500 a month. If the dealer applies the trade in as 10 months of payments, you're in good shape, your trade in just paid for 10 full months of your new lease. BUT, if the dealer lowers the price of your new vehicle and then recalculates your lease payments, you just paid $5k up front that you would have paid out over the length of the lease. The money is better in your pocket than the dealer.

It's different than when purchasing a new car. If the dealer gives you $5k for the trade in and lowers the purchase price you pay less interest and pay off the car earlier and when the loan is over, you own the car. On a lease, you're just paying more for the car and when the lease is up you don't own the car, you've just paid more for it than if you didn't put any money down.

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You have your way. I have my way. As for the right way, the correct way, and the only way, it does not exist.