I'd call that "web(site) development". Most degrees are modular, so it might also be possible to do a more general computing course, but specialize in web development after the first year, I'm not really sure.
I think web design is mainly graphic design, plus some css/html.
Learning the program would be the best course of action, and besides there's more money in web application development than just pure web design.
PHP/ASP/MYSQL ect. would all be easier to learn from there, IF the school hadn't taught it to you already.
I could also suggest buying some books and learning yourself.
This stuff is constantly evolving, anything you learned in school would most likely be obsolete be the time you finished.
well I've done 2 years of Computer Science studying Java. And I did a year college on Web design but like you said it only touched on HTML/CSS which I already know and the art side, where I've been using Photoshop for over 7 years it was quite boring to say the least of a course. I've taken 2 courses on Internet Marketing now, which cover Search Engine Optimization (SEO) and Search Engine Marketing (SEM) but I want to learn PHP, MYSQL etc to be able to create software like Wordpress, or Communities like PHPBB or Scaled down versions of things like Facebook or Myspace.
Yeah, I thought computer science would cover it, and the electives would let you specialize. Though what degrees really teach you is how to learn, so you should be able to figure it out yourself afterwards.
Disclaimer: Any sarcasm in my posts will not be mentioned as that would ruin the purpose. It is assumed that the reader is intelligent enough to tell the difference between what is sarcasm and what is not.