They did not write the more accurate headline:

"Trump backs down on Iran threat, Claims of Negotitations Denied by Tehran"

NYT tends to be excessively charitable with what Trump does and says. This is because it's more important for the NYT to use language which sounds neutral, rather than language which reflects reality. Also, for whatever reason, the usual obligation of journalists to decry falsehoods does not apply to things Trump says off the cuff.

It's not necessarily a bad business decision though, a paper which applied the same standards on Donald Trump as it does on everyone else would presumably be even more painful reading.