Production Stage Manager, Stage Manager, Assistant Stage Manager are terms used to described different members of the team - what the team members are assigned to due, is up to the team.
At my theatre, as PSM, I over see a show with ASM, Production Assistant (read that as a non-equity ASM) and a Stage Management Intern - on a typical show. But on the reps I am currently working on, I was PSM, over 2 equity SMs, 2 Production Assistant and 2 SM Interns.
I think your director has very specifics wants and/or needs for you as PSM, and you should sit down and ask what he expects. As PSM you may just be asked to oversee the entire production, ala a production manager, or he may expect you to be the lead SM. There are all different sorts of break downs - especially what production level you model yourself off.