The web design/development team at the company that employs me breaks down like this:
Project Manager The PM creates and monitors the schedule, staff & budget. It is their responsibility to schedule regular meetings, and make sure the client's needs are being met.
IA/User Experience Designer (yours truly) The IA lead works with our strategy team to determine site structure and user interface design. Content, functionality and usability testing are also part of the UXD's role.
Web Graphic Designer The designers are responsible only for the visual design based on wireframes created by the UXD. They have no coding responsibility, most had zero coding experience until I taught a CSS/HTML development course. They also create designs in Flash, but do no ActionScripting.
Front End Developer The Front End or UI Developer's job is to take the visual design from a PhotoShop file to a functional HTML/CSS template. Most have design skills, but they are not trained designers.
Back End/Database Developer These are the C#, ASP, .NET and Java developers who take the static HTML templates coded by the UI Developer and convert them into dynamic documents and websites.
I think that the Web Designer who has no ability to code HTML, CSS, and even JavaScript is a dying breed. I see the industry breaking into two distinctive skill sets: front end (UI, graphic design, Flash and HTML/CSS/JS coding) and back end (server-side scripting languages, databases). There are some people who are phenominal at the design and front/back end development aspects, but these people are few and far between (and most live in their parents' basements).
I like what adjustafresh added. I love the title Front End Developer and loathe, like many others out there, the title webmaster or web designer. Those two titles are associated with table based designs and very poor coding. I take pride in coding websites up to standards and don't wish to be thrown in with all the others out there who don't care a thing about it.
I absolutely agree with what medigerati said. I hate being put in the "web designer" bin, right now I call myself web designer/developer, but i like Front End Developer much better, as that is what I do. I design, conceptualize and develop according to standards, that is much more than just "designing" the website.
I guess we could just go back to the days of the "webmaster" though.