So by stigma you mean someone that adapts to changing times and learns many new technologies? I am down with that. I think the stigma is all in your head. Classic programming doesn't force you to branch out into other technologies (SQL, XML, web services, etc) as much as web development.
What is wrong with being a web developer in a time when more and more applications are being ported to a web-based setup? One of the many great things about being a web developer is you get to work with a very wide variety of technologies and other programming languages (Javascript, C#, VB, Java, Ruby, etc).
This would look great on a resume showing that you can pick up new programming languages and adapt to an ever-changing world instead of classic programmers that get stuck on the same programming language for decades. Go where the money is going, that is towards the web. The signs are there, desktop apps are on their way out and the world is becoming more and more cloud-based. The great thing about web development is you get to learn so many things that branching out into new apps and technologies will eventually become a breeze because you have already had to learn so much.
Also, if you use a language like PHP you don't have the frustrating issue of the stuttering debug method. Write code, save, compile, run code... problem found, trace bug, apply fix, save, recompile, run code... crap another issue, trace bug, apply fix, save, recompile, dang it a syntax error arrgg, fix code, save, recompile, test.....(you get my point). With PHP it is more like write code, save, test, found a bug, fix bug (tracing info comes with specific line numbers in errors, hooray), save, test, fix another thing, save, test, sweet it works!
I would think long and hard about becoming a stagnant desktop developer. Don't let the classical programmers confuse you!