Normally, you should enter the W-2, exactly how you see it. However, you should verify with your employer and get a corrected W-2.
If you made any money, claiming tax-exempt or not, the amount should be reported in box 1, 3, and 5. If you underreport your income, you will likely have issues with the IRS. It is normal for some W-2s not to have any entry in box 2, but not in box 1.
You can claim exempt on your w-4 for withholding exemptions from wages (which only effects the amount paid in, not your actual tax liability). Being truly tax-exempt is reserved for 'Employees of Foreign Governments / International Organizations' and a few other very small groups of individuals.