This piece describes the difference between network and exchange pretty well: http://www.adclarity.com/2014/10/ad-exchanges-vs-ad-networks/
Basically the difference is just who they serve. Networks were made for publishers, excchanges were tacked on to make life easier and cheaper for buyers.
I've never, ever heard of an ad mediator. I wouldn't get too hung up on the exact differences between an exchange and a network. If you are a small business trying to get started in the online advertising ecosystem, I would honestly start with google products. You're probably already using adwords for search, so it will be really easy to run display campaigns on there as well. These also hook up nicely with Google Analytics.
If at some point you want to run a larger budget, need more inventory, want different targeting, have complex ad serving needs, or want to run video, then you should probably talk to a freelancer or small agency for help. It's a pretty complex ecosystem.