It is not a correction you are 100% right. Each school has different COA's and the financial aid office at each school can tell you the grad plus award etc. At my school the max you can get I believe is 14,000 a semester for COA, but I take out 12k. The grad plus loans are not to difficult to obtain as long as you are current on any prior educational loans and have no judgments, bankruptcies, etc.
How it works to follow to the OP's comment is the loan will be a refund for the money that does not go to the school and whatever your schools COA for the semester is, which can probably range from 7-19k depending on location you will have that amount appear in your bank account.