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The non-religious are a category of people distinct from atheists. The non-religious do not subscribe to any particular religion, but see religion as a valid means of understanding the world. They do not disapprove of religion. This is in opposition to atheism, which sees religion as essentially a mistake.
Non-religious people may still have a cultural bias towards some religion; for example, American non-religious might be happier if their children become Christian monks rather than Muslim imams. But at some level they recognize this as a cultural judgment and do not raise it to the level of choosing spiritual truth over spiritual lies.
My family would probably be described as "non-religious." We went to church when I was little but my parents now allow us kids to visit with any religion we like. The laziness of parents allowing children to decide their own world view for themselves, rather than drilling into them the importance of rejecting all superstition or all science, is known as trust.