I've worked with a few people who have the word "webmaster" either as part of their formal job title or as their actual title.
The type of work these people do does relate loosely to the web, but I suspect better, more appropriate titles that more accurately reflect the job function would make more sense.
Is the "webmaster" moniker still relevant today?
I personally believe that the word "webmaster" is something vaguely defined.
It comes from the times when web programming was mostly about static HTML and a little bit of scripting. It was the time of web sites.
These days we are talking about web applications. Something with a heavy server side part, when the problems of data model design, application architecture, scalability, performance and maintenance play the key role, while HTML/CSS/JavaScript are only used as a tool to build UI.
I think today this term is replaced by more specific roles like web designer, usability expert, UI designer, web developer, DBA, system architect etc.
The term, "webmaster" was obviously created as an analogon to "postmaster", which is the standard email address for addressing email related to the email (SMTP) system itself. So it really is an email address, a contact point rather than a job description. However, few people realized this even when the term was introduced, so it's really been obsolete, in the sense you describe, ever since it started becoming popular.
You'd never know it from sites like Stackoverflow, but there are plenty of sites that are simple, plain old html, with maybe one "contact us" form. The person who writes and updates that kind of site isn't a programmer, an IT professional, etc. - he's just some guy, y'know?
Usually a Webmaster is the person responsible for the content on the site. The main person to contact regarding the site.
But as other jobs, what a webmaster does also can be varied. He can be the programmer of the site. Then he is taking on the web developer role. He could have designed the look of the site. Then he is taking on the web designer role. I think you get my drift.
I think webmaster is more of a broader skill set, possibly for a company that has a smaller load site. I.E. they can do the design graphics, the front end scripting and code as well as the back end. Most big companies now split these roles... But it doesn't mean somebody can't run the whole show themselves... I know many who can just "do it all". Some smaller shops might just want one person to know a little bit about everything. I think if you are a freelancer doing gigs, the term can still be relevant as you might have both designer AND developer skillset, plus run your own server to host, etc. I do think however, that Web Administrator would be a better modern term. Google still has a "webmaster" tools for instance. It is more of a generalist role, that can handle all of the other common functions such as web designer, web developer, web administrator. Some companies just don't have the resources to piecemeal everything together, so they hire one person who can do it all. Or have one person maintain their corp website. I do exactly the same thing often and if I can't handle the work I have contacts that are willing to take on certain pieces like the graphics or possibly set up a specific ecommerce app, etc. Even though I can do this myself... sometimes its more of a payoff to have somebody else do it.