Terminating car lease early, paying bank responsibilities

I currently have a car lease with two months left. A dealership has the next car I’d like, and will do a trade for my current lease. They will cut me a check for the remaining lease payments.

I called the bank that owns the lease to get info on end-of-lease requirements. They said in this scenario where the lease is being terminated early, there are two possibilities:

1. The dealership will buy the car from them outright. The bank is made whole and end of story for me.
2. The dealership will return the car to the bank, if they think they can’t sell it themselves for a good price. The bank will sell it themselves. If the sale price is less than what they estimated the value would be at lease end, I’m responsible for the difference.

The dealership tells me they probably will buy the car. They said in either option, I wouldn’t be responsible for anything else. Is there a form that they should be giving me that puts that in writing? If so, what should I be asking for?

Thanks