In other words, the graduate college / graduate school wants the scores, not necessarily the department. Is just one copy of the GRE sent to the "institution", and forwarded to the "department"? Or can I just leave the "institution" and forget the department, and expect that it will reach the graduate college of the school?

I had the same question yesterday. I emailed the program and asked them specifically for their department code just in case. I’m taking my GRE tomorrow and totally freaked out. Good luck!