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Regardless of your repeated attempts to define it as such, atheism is not a religion.
Except when it is.
The OP has hit upon a nice parallel. There are people who call themselves Buddhist, Confucian and Jewish who aren't religious. These things are often called religions because there are other people who call themselves Buddhist, Confucian or Jewish and who are religious. Those religious people use these labels to describe what religion they adhere to.
Yet, even when one of these words is used as a description of religious belief, each refers to a bundle of distinct religions, plural, not a religion, singular. An observant Jew may recognize as a fellow observant Jew someone who, for religious reasons, dresses differently, eats different foods, and keeps more or fewer holy days, with different customs for each of them.
If there is a criticism to be made, it is that the OP went too far afield from the usual religions discussed here to fetch an example. The same points could be made about
Christian.
I am a cultural Christian, my religion, however, is agnostic. There are plenty of cultural Christians. Used alone, though, the word typically means something or somebody religious. When it is used as a religious word,
Christian refers to thousands of distinct belief and ritual systems, not one singular religion.
As words go,
atheism has only relatively recently acquired the meaning of voluntarily having no godly traffic. It is obvious that the langauge hasn't yet completely sorted out this word's distinction from other forms of informed restraint in trafficking, like that described by the even newer word,
agnsoticism. We still don't have a word for those who would give only a non-responsive answer to the question of God.
Nevertheless, the outlines of usage are emerging, and they are just as the OP suggests. Although it isn't yet at the buzz level, "cultural atheist" makes perfect sense as a term. It makes perfect sense because of its obvious parallelism with "cultural Christian" and "cultural Jew." It isn't buzz yet, I think, because there aren't enough atheists for many other folks to care which ones are religious and which ones aren't. Among those whose religion is atheism, there are distinct viewpoints about religious questions. All that atheists, religious or otherwise, have in common is their answer to one question.