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. Once 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. It may ultimately simply 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.
A basic understanding of what Javascript/JQuery does is needed, although actual functionality would be something I'd not really cover were I teaching. Just the concept of dynamic pages via javascript should be understood. Functionality may require its own class or dedicated area of study and you start blurring the lines with the developer phase due to possible ajax interactions.
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.
Additional:
I look at any general educational experience as a multi-tiered experience. It is important to expose new students to the wide variety of possibilities, if only minimally. Then the students themselves can better target and move towards the areas they find intriguing and exciting. As a basic core, any student under the "graphic design" umbrella should be exposed to the basics of web design, print design, video/multimedia/game design, and marketing. I feel it is most likely these areas which will inspire and help students realize their own strengths and weaknesses. If a student then wants to focus more on web design, they should be able to seek courses teaching advanced CSS or jQuery/javascript. Just as if they wanted to focus on the marketing side they could seek courses in demographic studies and statistical analysis. Make no mistake a "Graphic Designer" with strong web skills is just as valuable as a "Graphic Designer" with strong marketing skills. However, in my experience, when the Human Resources department places an advertisement for a "Graphic Designer" they are seeking someone knowledgable in either print or web design and most often both to some degree.
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.