I guess it will depend on the complexity of the site.
If I created a small web site then I would be the webmaster responsible for everything - designing and creating the site, making the graphics, uploading the site, checking the site works, checking performance, checking security, checking if it appears in search engines, checking usability and so on.
But for a large sites like say Amazon or Facebook there will be dozens of webmasters, broken down in to different teams.
One team may design the site, another build it , another test it, another update it, another may look after the hardware, another look after the software, others may just look at security or ease of use etc.
So on a large site a "webmaster" may be a virtual person, and any problem with a web site could be passed on to one of a dozen or more people.