It does mean just that. The same reason why both (in the flawed view of the author) are ineligible for the vice presidency is that both (supposedly) are ineligible for the presidency. That's the stipulation in the 12th amendment that is said to apply to the both of them.
And, once again, Bill Clinton was the legal president, no matter what claims Hillary or you or Bill may make to the contrary. It was Bill Clinton who signed legislation and was CinC (shudder) and whose name appeared next to electors on the ballot.
You can't both literally read the 12th and play games with the meaning of the word "president."
So either the reasoning is false or precisely doesn't mean precisely. Neither is complementary to the premise of the author.