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#6: i can see what you're getting at, but my point was that all views are based on core beliefs, including atheist's. Every proof has a set of presuppositions without which the proof becomes invalid (unless its a circular proof in which case it is meaningless anyway).
Therefore the latter group of people is empty. There is nobody that refuses to believe everything that is not backed by logic, because their own core beliefs are by definition not backed by logic, yet they believe and put their faith in those. A person in this group would have to believe absolutely nothing in order to refuse to believe in their own core beliefs.
The only way a person can avoid having presuppositions based on faith is to base them on existing logic (which is itself based on the presuppositions). This makes the reasoning circular and meaningless.
In your second paragraph you say that a view backed by logic stops being a matter of faith, however since that logic is based on presuppositions which are believed to be true by faith, that view is indeed a matter of faith.
I hope this makes my point clearer.