That New York Times article has it wrong.
It says, "'The rules say three months validity from the time of entry. It's a little complicated because they routinely give you 90 days to stay. So you really need six months' validity on your passport,'. Ms. Sprague said.
So she admits to the three months but gets it wrong---it is three months validity from time of planned departure from the Schengen zone.
That is what it says on the State Department website for Schengen counties:
https://travel.state.gov/content/passports/en/go/schengen-fact-sheet.html
And that is consistent with the EU Regulation 610/2013, enacted in 2013.
http://www.refworld.org/pdfid/545b82c14.pdf
Open the PDF and scroll down to the bottom of Page 4 and top of Page 5.
Maybe it is possible for individual EU members to have a longer period; according to some information above, the State Department says 6 months for Italy ( but the OP says the Italy government site says 3 months). Seems like the Italy site should accurately state the rule.
But note that it is 3 months after departure, not entry. If one is planning on staying in the Schengen zone for 90 days then you do need six months left on your passport.
I agree with Sasha that the six-month advice ( which you will see in many articles in addition to the NYT one) is routinely given out of an abundance of caution. But one should always distinguish between what is required and what is recommended.