The basic problem with most definitions I see about web designing and developing on this post is that there is this basic assumption of this waterfall method of development where "designers" make the site pretty, (some code the html) while the developer makes it function with javascript and backend code. This is fundamentally wrong because it assumes both designers and developers are part of the website team. If you think of the software life cycle in a proper project/product/development team, the design team is not part o the development, but part of the product team.
Designing a website is ALOT more than making it look pretty, its about understanding user behavior as it support product business goals and specifications. A website design determines how a user interacts with the website and ultimately whether the features/functions/purpose of the website is serviced by the design or not. So alot of UIX development, a/b testing, product is discussed because you cant talk about design without talking about product. This impact is not felt on small websites with little traffic, but on larger websites every and any change is tracked and analyzed for performance impact on the product and ultimately how it impacts returns on finances.
On the other hand, one thing I dont like about "programmers" is this design is a byproduct of functionality attitude they take towards design, and saying that 'web designers' are front end programmers reponsible for the markup that "real programmers" dont want to be bothered with.
Bottom line, if your writing code you are programming and thus a programmer not a designer. You can certainly wear both hats, and I often do, but I dont consider designer and programming the same thing. Programmers who dismiss html/css/flash/javascript and front end developing as a simply makeup of the program they built are fundamentally missing the point of what they are making. There is 0 point to making applications that users cannot user easily and do not serve the purpose of the application's creation.
the design of the website is the consumer facing side of your application and if you pay no attention to that part of the application then the rest is irrelevant, there's a reason why no matter how smart you are or good at what you do, you clean up and wear formal clothing to a job interview, its about presentation.
The perfect examples of this is companies like Apple, imagine if apple's product designers said "ehh who care about touchscreen and scroll wheels.. up/down buttons and an enter button work just as well"
Alot of the decisions you make day to day with products you use subconciously have more to do with the product design that delivers the functionality you want in its easiest way to consume. And believe me that was not achieved by accident.