But what's most important to me is the customer's preconceptions of what it means and what they think I as a "web designer" actually do.
I've gave up on the customers MISconceptions years ago. I'm usually working with corporate people, or government people all of whom suffer from the same desire to stand out as a leader without actually exerting themseleves. "What I do" is done to please me. If I did it better than the last time, I'm pleased.
I'd like to educate them on my definition of design, but most won't be interested.
Why? I don't want to teach them jack about my job, and I don't want to learn squat about theirs. I think it works better that way.

In this industry, there's a lot of sites produced - and supposedly designed - by 'kid next door', or the need for a web site is met with the phrase "my brother's friend's nephew does computers in school, he can make a web site!" There's nothing wrong with that, until said kid realises he could do the same slap-dash job for other people and charge professional rates (or more).
( note: i'm not saying they all do a crap job or aren't pro's so don't get your knickers in a twist) .
In principle, I could not agree more.
If you were a small business, (potential client of mine), how could you tell the difference between me and the proverbial 'kid next door'? And would you care?
No. As a small business, I wouldn't care. I would want the biggest bang for my buck. It would have to be proven to me, unequivocally, that the more expensive website was a better deal than the kid-next-door's. Otherwise the price tag will be the only thing seen.
- dM