I don't really understand what you mean by core beliefs outside a religious context. If you mean a person's outlook on the world, on their level of tolerance, liberalism, how sociable they want to be etc. then I can see what you mean about a lack of logic. But these attributes are not determined by religion; there are intolerant people with religious faith just as there are intolerant atheists, and liberal people with faith just as there are liberal atheists. Although your belief system, religious or nay, obviously has an effect on your outlook it does not, it appears, determine it.

These core beliefs, not, as you say, based on logic, are not determined by a person's religious beliefs.

Do you mean instead a person's judgements on the world around them? Here is where I feel the argument lies for this article.

On their judgements of the world a religious person has the base belief 'God is', with varying levels of non-logic based beliefs emanating out from that belief (with differences in ideas on creation etc.). An atheist has no base belief but logic, relying on what is proven to them by logic and demonstration.

Yes, these people have 'faith' in what is demonstrated to them, but this phenomenological faith is a very different beast to religious faith, and your confusion of the two is unhelpful. Faith (logic) is logic and faith (religious) is not logical. This is the distinction the article makes.