There are different kinds of leases and it depends on the lease if the locomotive is repainted into the railroad's paint scheme. Here's how I understand it. I welcome corrections.
First, there is the long term lease of new locomotives through a finiancial institution. This is similar to the way you or I would lease an automobile. The railroad leases the locomotives for a period of time, such as 15 years. At that point, the railroad can either buy the locomotive or return it to the owner, who can then sell or lease it to someone else (sometimes the original railroad). During the initial lease, these locomotives are almost always painted and numbered for the railroad, and seem to be owned by the railroad (just like you would seem to be the owner of a leased car) even though they technically aren't. Sometimes older locomotives that are rebuilt and recapitalized as new are leased like this as well.
Used locomotives are often leased under shorter-term leases. These leases can be of any duration, from months to years. Depending on the length and terms of the lease, they may be painted for the company that owns them, or for longer leases they may be painted for the railroad. Occasionally the locomotive will be fully painted for the railroad, but will have the reporting marks of the leasing company in small lettering near the number. If neither the leasing company nor the new railroad cares enough to paint the locomotive, it can remain in the colors of its previous operator, often, but not always, with the old roadname painted out. Sometimes a railroad will end up with a short-term lease on a locomotive it previously retired, and I think this is occasionally arranged ahead of time. Rebuilt locomotives can also fall under these leases. Many short lines have locomotives under these types of leases.
There are also "power-by-the-hour" arrangements. The Oakway SD60s and LMX B39-8s were originally examples of these. I'm not entirely sure how it worked, but I think in the case of the SD60s they were owned by EMD and leased to Oakway. Burlington Northern then paid Oakway for the use of the locomotives, but only when they were actually using them on a train. The LMX units on BN were in a similar arrangement with GE. The LMXs were sold off. Oakways are still occasionally seen on BNSF, but I don't know if the arrangement is the same. In this type of arrangement, the locomotives are typically painted in their owner's colors, not the railroad's.
As for the BN "whiteface" scheme, It was just a change to the cascade green scheme to make it more visible. I don't think it ever officially meant that a locomotive was leased, although at the time it was introduced, BN did receive a large number of rebuilt 4-axle EMDs under various lease agreements, so maybe that's why you thought that, John. Once the scheme was introduced, it was applied to any locomotive that was due for repainting, and BN repainted a lot of locomotives that they owned outright into the "whiteface" scheme. Naturally, the repainting took time, and there were many yet to be done when the BNSF merger occured.