From personal experience, many people say they believe there is "a higher power" because they have grown up with an established religion but realise the scriptures are man-made stories, but have not yet made the intellectual jump to deny the existence of a god of some kind.
Religion is unimportant to these people and they don't give it a lot of thought. They don't think it through and need a catalyst to go the final step to atheism or agnosticism. The trigger could be a book such as The God Delusion, a TV programme, discussing the topic with someone.
Dr Zuckerman suggests other reasons: bad things happening in life, other cultures with wildly different beliefs.
So, in surveys where people say they don't see themselves in any of the recognised religions but believe there is a "higher power", we can probably reckon on many of these people becoming fully-fledged atheists at some point.
You are pretty much describing my own journey to non-belief.