The official answer is 'no.'
It does not count as a 'strike' if a vice-president is elevated to president. Ford and Johnson have been the only cases since the passing of the 22nd Amendment.
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Section 1. 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. But this article shall not apply to any person holding the office of President when this article was proposed by the Congress, and shall not prevent any person who may be holding the office of President, or acting as President, during the term within which this article becomes operative from holding the office of President or acting as President during the remainder of such term.
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Johnson was allowed to run for a '
third term' since he served less than two years of Kennedy's term but declined the nomination because of the growing pressures of Vietnam. It is not possible that Clinton could run as vice-president.