That could open a tin of worms! The designer/developer thing has been discussed many time in the past, and there are a few different viewpoints as to what is what, trouble is no-one is really right or wrong, it's similar the "what is art?" question.
In recent years web design and development has has become a set of specialist roles including (a very broad/simplistic description):
UI designers (User Interface, ie how a button is designed)
UX designers (User Experience. ie how the button behaves when the visitor uses it)
Front end developers (HTML/CSS/JS, ie creating the button and functions using HTML, CSS and JS)
Backend developers/Programmers (PHP, ASP, Databases, make the site do stuff when the button is clicked)
Content and marketing (writing copy and promoting the site so people click the button)
... to name a few.