I'm not so sure:
Amendment XXIII read that to mean that the amendment didn't apply to Truman in any way.
...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.
I did in fact consider the the 22nd Amendment, which is why I left Eisenhower, Reagan and Clinton off the list.
Further evidence of that being the line of thinking is the second part of that sentence; the allowance of one who would suddenly be violating the Constitution upon ratification to finish serving the term. Never mind that Truman was previously exempted, or that thanks to the timing and the presence of the 7-year time limit, it could not have been used by anyone else.