I'm not sure you can safely make a objective distinction between "modern" science and what you're calling "science as actual personal knowledge". Science has a standard definition, then we all just add whatever baggage we want to it so that it fits our worldview. If you're delineation is subjective, it doesn't hold any water considering my point of view is different and there are thousands of other points of view different than either of ours.

Psychic phenomena occurrences are illogical according to empirical science, but they're not impossible. This would be a category that researchers would just label as "we're not really sure". That category exists in all major branches of science, so it's not scientific to say that supernatural occurrences are impossible. Illogical, sure. But logic has a dynamic definition that changes with technology, philosophy, theology, and many other "ology"'s. That being said, there is no such thing as a non-modern science that holds all of the logic. There's just your opinion based on what you consider logical based on a definition of science that you have adopted to suit your personal tastes. This is essentially what many people do to varying degrees. The real challenge is to try and push through bias and seriously consider all applicable angles to our life and universe.

There are qualified and degree'd scientists that believe in God. They guy who mapped the human genome believes in God. I would say these believing scientists have found a level playing field with logic and God on the same team if they're that entrenched and didn't throw out supernatural possibilities. Science does not say God doesn't exist.