I worked in one of the smaller branches for a few months as well, and it was nothing like central. Everyone knew each other, by face and name, and helping out in each others departments was a lot easier because there was only one or two people in each group. Having the adult reference librarian "in charge" happened more often as well because they were next in line in the hierarchy. So when the manager had to leave, it was usually the adult reference librarian who ran the library until the manager came back.
So all I did was Adult Reference. I did programs, worked on committees, created staff genre training classes, and had a few other non-public related duties; but, the majority of my time was assisting adults on the reference desk, or over the telephone, or by email. Now that I am looking for another job, it seems like that experience isn't enough. A lot of jobs I'm seeing want supervisory experience. Have I been in charge of other people? Yes. Many times. Was it in an official capacity? No, because that's what the volunteer services department was for, or the reference manager, or whoever had the "I'm in charge" phone that day. The same goes with working with vendors and budgets. Another department handled that. I was just a librarian.