I realize that there are misconceptions about this subject so let me clear this up. This is how things are in regards to atheism.
Atheism as defined in my Cassell's dictionary means:
"1 disbelief in the existence of God or gods. 2 godlessness, wickedness; disregard of God and God's Laws. atheist n. a person who disbelieves, or denies the existence of a God. "
As an atheist I can have my own spiritual beliefs without including a God or gods. The definition of god in my dictionary is:
"1 a superhuman or supernatural being regarded as controlling natural forces and human destinies and worshipped or propitiated by humans. 2 a deity, a divinity 3 (God) in monotheistic religions, the Supreme Being, the self-existent and eternal creator and ruler of the univers. 4 a personification of the forces of nature..."
If god was talked about in only a spiritual sort of way, without being personified and or/humanized in any way, I would use the word. I do not though. God got dirty when it started making people wealthy. You can buy a Jesus statue at the dollar store ... that's pretty bad in my books for many different reasons.
So I am an atheist for now, who does not believe in God and/or gods but who has personal spiritual beliefs. It works for me and the dictionary even allows it. lol
The answer to the first question I suppose would be a spiritual atheist, if it had to be specified. I do not believe there are churches for atheists. They would probably be split up depending on their personal spiritual beliefs.