Joe, you do realise you can cut the ends of the strings off, right? Generally when I'm stringing my guitar I'll end up with about 15-20cm of each string that I'll cut. You don't need to wrap the whole thing around the pegs. In fact it's better if you don't, an evenly wrapped (so it looks like a slinky, with no wind overlapping any others) stringing with maybe 3-4 winds around each the peg will give much better tuning stability (something I seem to forget every single time I put on a G string. Must be too busy coming up with bad jokes).
The one your guitar teacher put on probably snapped because of a sharp bit of metal on the bridge or something similar, if it just snapped when you weren't doing anything with it.
edit - in light of the pics, that's your problem. Thread the string through the hole in the peg so you've got say 15cm hanging out the other side, then wind. Just watch out for sharp string ends poking you in the eye while you're doing it. Just before you start winding, if you hold the string in place at the peg, a good measure is (as dakine said) that you should be able to get the width of your hand under it, or maybe a bit less for the thicker strings.