Is "Website Building" is right term?
Seems the perfect word for it, as it is a combination of 'Web Designing' and 'Web Developing' while also including other things related to it. I see it used often as a way to describe doing both.
If you're specifically talking about code (markup and and programming), you might as well use 'Coding a website' as building a website may also mean using a drag-and-drop tool of some sort.