Well, there are so many different things to consider in web development. You have user interface design and graphic design, programming, project management, content writing, etc. To be a "web designer" and create entire websites by yourself (without a team) you would pretty much need to have at least a little skill in each of these areas, and probably a few more.
Many of the large web development firms have teams of 5-7 people who have specialized tasks. That works really well for large sites, but is impractical for smaller ones. Depending on the market you wish to target, you could either be a 'jack of all trades' or specialize in one field and get really good.
I would have to say that a web designer is a person who had artisitic skills and the ability to express them through the mouse, in the form of a digital creation that represents a person, or an organization on the internet.