Another possibility is that the NYT edited/changed the article after the OP posted it. Ethics would dictate that they memorialize the change with a note at the end, which does not appear, but then this is the NYT.
The NYT changed their article and removed "Mr. Cruz said he would have no problem if one of his daughters was gay." The NYT did not acknowledge their error. They just made the change. The damage was done, of course. The NYT, IMO, knew exactly what they were doing, lied, and then changed their article after the lie had been repeated in countless blogs. Mission accomplished from their perspective.