Well web development is a broad term. I usually call myself a software engineer because that's what I do -- I don't do web design, I don't do SEO optimization etc. I'm purely writing the software. It just happens to run on the web! So just make sure the "web dev" course is actually what you want. If all they do is teach you some HTML, then it's probably not worth your time.
But the lines are getting blurred. Even if you write desktop software, chances are that you're gonna be interacting with the cloud somehow. So if you do any development at all you're gonna be learning the whole range of skills anyway. And depending on your technology stack (Java? .NET?) you'll be able to easily transition to writing web apps if you wanted to.
I could give you some free space on my server if you'd like. It doesn't have any control panels or anything though, you'd just get a db and then SFTP access to upload files.