As someone who wears all 3 hats, print/web/graphic designer, front-end developer, and back-end developer, I generally apply to whichever description seems the most relevant to my skill-set and what I would LIKE to be doing at the company(ie. which I would come off strongest in during an interview).
I've seen many job postings where Front-End Developers are just HTML/CSS luckily this seems to be changing to include scripting languages, which I favor. Generally the database, server work is handled by the Back-End Developers and with communication the business logic is handled by both but more commonly by the back-end developer or general programming guys. (Depending on company size of course)
Design belongs to designers (but really, more often than not, upper management). At this stage in the game, good designers can give you HTML/CSS, great designers can give you scripting work as-well. But their are also great designers who can't write a line of HTML/CSS, but that's more of a experience/working as-a-team thing. I would much prefer to work with a designer who can't code, but understands basic limitations of the web and is a fantastic mockup/layout-guy, then I would with a semi-decent mockup and semi-decent coding guy. (Let's face it, if you have time and it doesn't match your usual style, you generally try to make it conform as best as possible)
Anyways, like all answers...it depends.
Keep in mind too if an ad is looking for just a HTML/CSS guy; knowing how to script may be a bonus and may come off as such, until you sit-down and the 'coder' already in place, doesn't want you touching anything. Oh, silly politics.