When are the education tax credits applied? The year that you paid tuition or the year that you attended class?

I was in grad school from May 2012 to May 2013. My final semester was Jan - May 2013, but the tuition bill was paid in December 2012. I got a 1098-T for 2012 that had box 7 checked (the amount paid included tuition for a semester that started in January 2013). I entered all my tuition on the 2012 tax credit form, but it was capped at $10k for the non refundable education credits (less than 25% of what I paid) so I didn't get the full benefit. When I filed my 2013 taxes I didn't think about it and didn't take any tuition tax credits.

But now I'm wondering if I am allowed to apply the winter semester 2013 tuition (that I paid in Dec 2012) towards 2013 tax credits. If I can, I'll file an amended return and get some extra coin in my pocket. Doesn't change 2012 at all because I had maxed those out, it would just add more tax credits to 2013.

The tax credit form seems to indicate it is ok - Part III line 22 says you need a 2013 1098-T OR a 2012 1098-T with box 7 checked (which I have) to complete the worksheet. But then the instructions for that sheet say:

The credits are based on the amount of adjusted qualified education expenses paid for the student in 2013 for academic periods beginning in 2013 or beginning in the first 3 months of 2014

which seems to say that you had to pay the expenses in 2013 to get the 2013 tax benefit.

Anybody know?