The correct response to the question "Will the usability of a website be a part of UI or UX?" is:
The usability of a website is a part of UX design.
While usability can also relate to UI ( user interface) in terms of how easy and intuitive the interface elements are to use, it is primarily a core component of user experience (UX) design. UX encompasses all aspects of the end-user's interaction with the website, including usability, which focuses on how effectively users can achieve their goals on the website.