The path to Internet connectivity can sometimes be a slippery slope. It's amazing the kinds of myths you'll hear when the salesperson thinks you don't know anything about Internet connectivity.
Bits travel just as well over fiber, copper, or radio waves.
Since most local ISPs buy service from bigger ISPs above them, the speed of your connection to the bigger ISP is crucial to total performance. Despite what some ISPs might say, copper and fiber are a lot alike. Both copper and fiber can be wired redundantly so that a single cable cut doesn't take down the line. The only time you really need to have fiber is when you want a single circuit at speeds of 45 Mbps, usually called a T-3, or higher. Nevertheless, fiber sounds cooler and faster and better, so everyone wants it.