I'm going to answer your specific questions at the end.

I need more information.  What is your age?  This is a big factor in determining eligibility.

The statements do not contradict.  If you are eligible to be a dependent on a return, you can not claim the credit.  If you are not a dependent, you can claim the credit for yourself, your spouse or a dependent listed on  your return.

Look at this link and scroll down to the instructions for line 7: Instructions for Form 8863

I suspect you are under age 24 and a full time student.  This is usually when this comes up.  If your earned income did not equate to more than one-half of your support, you would not qualify.

For your first question - we can help you determine your eligibility when we have all the facts, so there is not need to claim a credit you aren't entitled to.  For your second question, if you do not qualify for the American Opportunity Credit, many times you qualify for the Lifetime Learning Credit, however, they are very different credits with different rules.  Let's see if you qualify for the AOTC first.

The fact that you are asking that question infers that  you are or can be claimed as a dependent.  As such, you are not eligible for either the   American opportunity tax credit or Lifetime Learning Credit or the Tuition and Fees deduction.

Furthermore, a  full time unmarried student, under age 24, is only eligible for the refundable portion of the American Opportunity Credit if he supports himself by working. You cannot be supporting yourself on parental support, 529 plans or student loans & grants. You usually must have actually paid tuition, not had it paid by scholarships & grants.  It is usually best if the parent claims that credit. 

You cannot claim a credit if you are, or can be, claimed as a dependent by someone else.

If you are eligible, you have answered  something wrong, in the interview. But, a lot of people are just not eligible. See https://www.irs.gov/Individuals/AOTC