To be clear, I'm talking about how someone defines themselves and their profession. Be a "software developer" or a "software designer". Don't put yourself in a "web" box, because the web will be implemented differently in a few years...like software always is.
There's a professional skillset that developers and designers should have that is far deeper and more general than this year's popular web technologies. If you have that skillset, your career is a lot more secure.
If you're a software developer/designer, and you think of yourself as a "web developer/designer", you're selling yourself short. And if you're beginning your career, know that you'll need learn how to develop/design software, not just "web".
Might be different in different regions and sizes of business - I don't work for a large business, I run a small business. I have called myself a Web Developer (if anything; usually I don't give myself a title) since 1995 and got by OK. There has been more than enough work for 15+ years. If the technology changes a bit, we either adjust or we don't, right?