Hello Kaileigh. To answer your first question, in many socieities "Wizards", or "Magicians." were seen as those who could do what others couldn't. In many tales the King/Queen of a kingdom would call upon a wizard to grant a specific request that a normal person couldn't achieve. Wizards were normally called Mages, the title Wizard comes up in a lot of Roman tales of how a wizard gave the valiant warrior strengeth to defeat the dragon and save the princess etc. So yes I believe Wizards were seen as the High Mages of society, to help mantain balance, therefore steping in when needed and to educate the non- believers.
Magicians however was a name or title, that you would hear belonging to a person who does things along the lines as a magic act. Wasn't a very common title back then, mage, or a wizard was pretty acurate.
I couldn't 100% tell you what a Wizard would seek considering I'm not one, nor do I call myself one. Though if I was to predict or assume, is that Wizards were interested in learning new spells, keeping their craft a secret. I cant give you a complete accurate definition on that.
I don't particularily know what you mean by, does magic work? Magic is just a form of energy that every living being possess (As far as we know)) Magic, isn't waving a wand around and shouting out an incantation, but being able to connect with your inner spiritual energy and the energy around you.
I hope what I said will be able to help you a little, I won't go further in depth with my last part, just to stay on topic.