I agree, the UI does not explain what the difference is between “Icon” and “Image”.
But if I would have to venture a guess, an icon simply means the tiny image, and the image means when you look at the badge’s page you get this fancy image … but I would have to look at the source to know what the difference is for sure.
As for the PS
Unfortunately, I haven’t managed to get links to appear properly in the description fields on the settings page, but if you’re looking for FA icon classes then look no further than http://fontawesome.io/icons/