Hi there,
I'll make an attempt to answer your question based on my experience (I've been in the webmaster-type area for around two years now, so I'm still relatively green
First, just to commenting on your 2 points:
1. Working the website - In my role, I'm responsible for all the coding and design work involved, but the actual content comes from the various groups within my organisation. Because of this, I'm acting as a liaison with these groups to obtain and work on content to ensure such content is kept up to date, and is a form that works well on the web (in terms of readability/useability). In addition to this, I also need to manage the webs logical structure, and report on usage trends.
I'm also responsible for evaluating the technical feasability of any new developments and web-applications that are proposed, delivering plans and strategies on how to implement such proposals, and then delivering a solution.
Another large part of my role in "working the site" is education of the organisations staff in terms of what is and is not acheivable within our site (being a government department, we have responsibilities in terms of useability and some proposals have the potential to become useability nightmares.)
2. Increasing traffic: I'm of the view that it is not solely the webmasters role to increase traffic to a site - I feel that marketing groups should headup this area, with the webmaster providing technical support and to implement marketing strategies in the web area.
Further to those two points, I also have a degree of responsibility for the web server, although much of that area is managed by our server management staff.
Finally, I act as a liaison to outside organisations/groups/public users to provide support when problems are encountered, and to maintain (share) links and information (including development strategies, experiences regarding certain projects etc)
As I said, this is based on my experiences within one organisation - I understand responsibilities will vary depending on organisation types, structures etc.
Nonetheless, I hope this helps.
Dave