I see web design as a subset of graphic design and not always a separate entity so I struggle a bit with the "vs." aspect of the question edit. One can certainly only be a web designer. I'm just not sure focusing so directly on one area makes that person a "graphic" designer. Much the same way one can be a fantastic package designer, but may not really have the skill set to qualify them as a "graphic" designer. I do feel, however, that "Graphic" designer encompasses both print and web reproduction. Therefore the "vs" doesn't play well in my mind. It ultimately, may just be semantics.
Note: This response was prior to the edit of the actual question being asked:
I've been doing both print design and front end web design for a couple decades. I build web sites with a variety of HTML, CSS, jQuery, PHP, MySQL, and when forced, ASP. I do not consider myself a web developer. I am strictly a designer.
I feel, any good designer needs to understand HTML/CSS, Spot Color, RGB, CMYK, resolutions, and separations. This covers the basics of web and print reproduction.
It's important to understand the restrictions when designing. Without an understanding of how separations work, print design can suffer. The same is true for web design - without an understanding of how HTML and CSS work, web design can suffer.
As I'm sure you're aware, web building is really 2 stages - design and development. The more those stages can blur for either the designer or the developer, the better the end result can be. A designer who understands there are restrictions in how HTML and CSS work will not be creating designs that are near impossible to be coded. In the end, I look at HTML/CSS as necessary core knowledge in today's world. Face it, HTML has been basically the same for more than 15 years with only minor updates. And it can be learned in a week or less of simply experimenting. There's no reason a designer should not know it.
I don't think a designer needs to necessarily understand any PHP/MySQL or other server-side scripting/database systems, although it can certainly help. The simplest thing such as a PHP include can drastically reduce the time needed to flesh out a series of web pages.
As for the ads that want everything... They ask for the world realizing no one actually possesses all those skills. But if you leave something off, you may not get that one application from that one candidate with one spectacular skill in that one area. It's a grab bag. No one should feel inferior or insufficient because they only possess a third or three fourths of the listed skills in a classified ad. Most placing those ads are aware that no one can do it all.
I chose to go in the direction of print, and then subsequently web front-end. But I could have easily chose to go toward print and marketing or video and then web. Without the initial exposure to each area I never would have known where my skill set (and passions) lay.