You were both on the right track. The Twenty-Second Amendment says
No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once.
Nixon was elected twice, so he can't be elected a third time. However, if you're, say, the VP for two or more years of a term and then the Prez bites it, you can take over for the remaining two or fewer years of his term and then be elected for two more terms. It doesn't apply to Nixon, but it's something.
I'm just like Teral! I'm just like--! (spine breaks with a deafening crack