I think that the key to the answer is one word that you have in your question: "artistic." I have worked with a lot of artistic types who could not design a website if their life depended on it – some would react to the idea of web standards and patterns with disdain or disqust. I have also worked with artists who look at the medium of web design as a palette to explore and create in.
So, do you have to be "artistic?" Nope. You do have to be (if you want to succeed) focused on solving problems for your users and supporting the needs of the business and understanding the constraints. The best web designers are creative and "balance" all the conventions and constraints to create an exceptional solution.