Changing my name shown atop iTunes 12 (eg from 'Thomas' to 'Tom')
I wanted to do this too (I changed my first name), and stumbled on a solution somewhat accidentally:

First, I changed my Apple ID first name at http://appleid.apple.com/ (as you already have).

Next (wondering whether or not the name change had taken effect anywhere), I went to look at my account information. It should ask you to log in. For me, that is when iTunes figured out my name had changed; it updated immediately.

Based on this, my guess is iTunes just doesn't bother to check whether or not any details surrounding your Apple ID have changed except for at log in, so when you do that, it should just update.