There seems to be a common misconception among religious people of all kinds that atheists are somehow less moral, and that religion "enriches" one's morals. And so moral and religious discussions are often intersected. I think this is wrong in more ways than one, but for the sake of character limit, I'll save it for now.
The question is how do we separate the two? And this is what the idea is about - we need a new term beside "atheist" to serve (from the viewpoint of religious people) as a middleway between religious (and therefore moral in their view) and atheist (and therefore immoral in their view). This term however must not be exclusive from religion... it needs to be inclusive.
So, without further ado, here's what I believe is the best such term:
Moralist - Anyone who follows the "golden rule", including in cases of conflicts with holy scripture.
The conflict part is critical. With it, you label any non-fundamentalist religious person AND the larger part of atheists.