Re: Is Trump POTUS or not POTUS?

The Yell wrote:

"The first inauguration of Calvin Coolidge as the 30th President of the United States was held on August 3, 1923 at the Coolidge Homestead in Plymouth Notch, Vermont, following the death of President Warren G. Harding the previous evening.

Isn't that example when a vice president takes the office?  That's a little different than a hand-over between presidents representing two different parties.