That's a good point, and I think it points out the fundamental misunderstanding in the comment I was replying to: conflation of atheism with radical atheism, particularly New Atheism.
My comment didn't mention 'radical' or 'New'. The location of "belief" makes the sentence an accurate description of atheism or not.
Adding 'types' in front of the word just continues us down the path of words not really meaning anything anymore.
Are radical atheists just atheists that act like dicks about it? Don't call them radical atheists, just call them dicks. Is a New Atheist one that does interpretive dance to explain their belief that there is no god? That's just an unlikely combination of two things, one of which is atheism.
Eg. I'm a radical grammarian (a dick when it comes to grammar).