Forum Moderators: open
Forget what unrelated others might think of you, that is mere vanity.
What do your clients think, that matters.
That leaves a lot of people as web developers. This usually means someone who knows how to install Wordpress and plugins and install a drag and drop theme.
Web stuff no longer even has the benefit of being new and exciting. Its now old and boring and the hot stuff (in the minds of business and the public) is social media, SaaS, and cloud in general.
web developer ... usually means someone who knows how to install Wordpress
web designer ...will draw a concept for a website in Photoshop
Writing content for websites is likely to get you regarded as interchangeable with people who write "SEO content"
no one uses the description webmaster except in this forum
The future lies in DIY sites for simple cases and SaaS for everything else.
Work that I don't do but what they typically don't need need is an elaborate web-application.
But a true webmaster will still outrank them in the SERPs.
I would describe a webmaster as someone who does all of the above and more, including graphics, multimedia and site maintenance.
Is it better to specialise than risk being a jack of all trades?
I don't believe one has to know it all to describe oneself as a webmaster. There are lots of specialised skills you arguably would be better buying in rather than making a perhaps poor attempt to learn. Good to know an area I think and then buy in the rest from trusted parties as and when needed. That is certainly what I used to do when I was responsible for developing (smaller) websites.