Website design includes more abstract elements such as usability, ergonomics, layout traditions, user habits, navigation logic and other things that simplify the using of websites and help to find information faster.
Graphic design is the creation of new solutions to problems within the medium of visual communication, either in 2D or 3D, on paper or digitally. The output is usually something that can be reproduced in order to deliver its objective many times.
Graphic design output includes logos, web site user interfaces, tattoos, sketches for fashion or industrial design, typefaces, print, animated characters, advertisements, flyers, movie credits, newspaper headlines, and designs for textiles. This is just a small part of what can be designed graphically. What they have in common is output that will be consumed through visual media.
Web design is the creation of a new solution to a problem in the web medium. It includes graphic design, because the output will be consumed through a visual medium. But it goes further than the purely visual. It stretches to incorporate every aspect of a web experience.
For example, choosing the right text contents for a web page is part of web design. This text may be viewed on search engine results, which is part of the user experience. And the position in the rankings of that result is also part of the experience, so I would argue that search engine optimisation is part of web design also.
Designing a web experience may incorporate some or all of the following list of disciplines. Some come under graphic design, others are broader design.
The bottom line is that all these skills and activities can impact the success of a web site and the experience of using that site. This shows the breadth of the web designer’s toolkit. Anyone who only offers the graphic aspects is not a web designer. They’re a graphic designer who is doing graphic design for web pages. And there’s nothing wrong with that. But do not trust a person with only graphic skills to build your web site, and it will not be anywhere near as successful as it could be.