Great post, I have a similar but slightly different view:
Web site owners usually have a goal that they want to achieve with their site. Like customer relations, provide information, sell stuff. To design an web site and therefore help the web site owner to reach his goal, you need to create a site that looks great and that visitors use the way you want.
These to things require a completely different skill set and techniques to master. To make your site look great, creative, attractive, etc. you need artistic and creative design skill, good photoshop and css knowledge. To make a site work functionally, you need to understand how visitors are using the site, you need to be good with analytics and stuff.
When a web site owner comes to you and says “my site isn’t working, do something about it”, its either because the site look dull or doesn’t work structurally. Depending on which is the case, you need to redesign or realign. If the site fails on both subjects, I would call it rebuilding (=redesigning+realigning).