What counts as a term as President of the United States? When the Vice President takes over after an assassination or resignation, does that count as a term? If the VP then decided to run independently as president, would he then be limited to only serve a single full 4-year term in office?
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 President more than once.
From: 22nd Amendment
So that somewhat answers what is considered a term length. If you serve as president for more than 2 years, at least in terms of re-election, that counts as one full term.
That same page then goes on to explain how many years a president may serve:
However, it is possible for an individual to serve up to ten years as president. The amendment specifies that if a vice president or other successor takes over for a president—who, for whatever reason, cannot fulfill the term—and serves two years or less of the former president’s term, the new president may serve for two full four-year terms.
One's status of VP has nothing to do with one's term limits as President. They are separate positions and aren't tied together in terms of term limitations.