New Atheism had no founder, no organisation, no membership and no structure, and any account that supplies one is describing something that did not exist. It was a label applied to a run of bestselling books, coined by the journalist Gary Wolf in a magazine article in 2006, and it named a mood rather than a body.
The mood was this: that religion is not merely false but harmful; that treating religious claims with more deference than other claims is a category error; and that politeness on the subject had become a form of collusion. Earlier unbelief had generally been private, or philosophical, or content to leave the religious alone. This was not.
Four books did most of the work. Sam Harris, The End of Faith, 2004, written in direct response to the attacks of September 2001. Daniel Dennett, Breaking the Spell, 2006. Richard Dawkins, The God Delusion, 2006, the commercial peak. Christopher Hitchens, God Is Not Great, 2007, subtitled with the claim that religion poisons everything. A filmed conversation among the four in 2007 was released as The Four Horsemen, and the name stuck to them.
The timing is not incidental, and the participants did not pretend otherwise. A movement arguing that religious belief has consequences found its audience in the years after religiously motivated mass murder, and during a period of religious mobilisation in American politics.
None of the four founded anything, and none of them asked to be grouped. The label was applied by a journalist and adopted by an audience, which is the ordinary way such currents get named.