Does Cap Cost Reduction Really Save On Interest?

Based on the article “How To Calculate Lease Payments By Hand”, the monthly interest can be determined by:
(Capitalized Cost + Residual)*Money Factor

Based on that formula and nearly everything else I’ve read (but perhaps misunderstood), it would seem that anything that reduces the capitalized cost (such as a trade-in) should reduce the interest amount. But then I came across this article that says otherwise. Here is a paragraph from the article.

A cap cost reduction (via trade-in or otherwise) on a lease won’t save you any money on interest, the way a down payment does when you finance a car. Interest charges on a typical contract are locked in, regardless of whether or not you put any cash up front. However, the fact that it can lower your payments at least seems helpful to many consumers.

It seems that this company originates auto loans and has an interest in encouraging people not to lease. Are they incorrect? Or am i missing something basic about lease calculations?