Legal name is the one that appears on official documents, like birth certificate, unless you have legally changed it. This is important to the school because it links to things like SSN.
Preferred name is the one you go by. Teachers will probably use your preferred name when they refer to you in recs as that is presumably how they know you.
At the school I am most familiar with (George in PA), nobody at the school would use any name other than your preferred name.