By soul in this article, I mean a supernatural extra object, but I am aware that many people here rationally reject that notion. What I was trying to get at was that even though we understand that it is not true, many hidden thought processes take the existence of a (supernatural) soul for granted.

I am curious about the opinions of other people here about what actual physical processes would comprise a non-supernatural soul. If I replaced all of my insides with such advanced electronics that nobody else would notice a difference (without a medical examination), does that mean I have the same soul? If I had to define a soul it would be "whatever it takes internally (mentally and physically) to cause an identical outward behavior as observed in a certain point in time". That satisfies the conditions of your definition, as well, although it is not restricted to "people", but actually comes closer to defining what qualia actually are.