Your first translation was what I thought at first but then had it explained. It means that you are in charge of something, for example during the party you are responsible for the drinks, someone else - for the music, etc. It doesn't mean that you are a responsible person, it also does not ask if it is your fault.
Zuständig has the same double meaning it does in English, of meaning either that someone is responsible as a general character trait or that they are responsible for some specific task. The difference in verantwortlich might be easier to remember if you more literally translate it as "answerable for" -- it means something is someone's fault, usually something bad.