first of all, i'm glad you got your puch to start and that this website (and fred's most excellent guide) helped you out.
on to the mirrors ... as far as i know, you should be able to just pull the mirror out. i don't use mirrors, and don't especially like the look of that kind (i used to have the kind that mount on the handlebars themselves), but i think that kind is like a cork, you just "plug" them in and then tighten them to prevent them from spinning freely. now that you loosened it so that it turns freely, you should be able to just pull it out.
the reason i don't like that kind of mirror is that it requires you to rip a whole on your handlebar grip. if you want another mirror option, get the kind that clamps onto the handlebars. you can place them at different angles/positions and they are a bit cooler looking, in my opinion.
btw, you have a nice, sharp puch. oh, and the company is still around, from what i can tell. it's part of the daimler group and is based in austria. mostly, the mopeds you can get in the us are hero-puch, which come from india. but i'm very sure that puch is still making mopeds for the european market.