I do understand the terms of a basic lease; it’s the buyout on top of it that is a mystery to me. I also have never tried to negotiate beyond comparing things like incentives and discounts between dealers.
There are a lot of discussions about buyouts happening, but not a lot of details. For example, many threads (and yourself) have said “just call the bank after and negotiate a buyout.” Negotiate how? Do I offer a price and hope they accept it? Is it part of the lease agreement? What can I do or not do? I have no idea. Unlike with a lease which is well documented to a T, complete with entire YouTube videos, the actual process of negotiating an immediate buyout seems much less documented.
Many of the topics from @jeisensc search above give disjointed information; for example, this one with Chrysler mention the credit being applied to the RV. According to this topic, Lexus Financial Services requires you to go through a dealer to buyout (not sure if that’s a state or manufacturer requirement). In another case, it looks like they used their own bank, Ally, to lease the car, resulting in a much clearer handoff. There’s also just very little data on Kia. 
So most of the time, I don’t know what’s state-specific, brand specific, dealer specific, etc..