"Any label you choose is fine by me. I'm just asking for some example of evidence clearly indicating a non-natural explanation. And, as I also asked, the means by which this non-natural explanation was derived?"
I have not researched other peoples methods for determining the supernaturality of something.
But off the top of my head, the logical method is as fallows:
If natural means fail to explain observed phenomena that is believed to be natural, there is either a natural aspect of the phenomena that has not been observed, or the phenomena is supernatural.
If all aspects of the phenomena that can be observed, have been observed, and there is still no natural explanation, any further attempt to explain it by natural means is pure conjecture.
There are many things that point to a non-natural 'explanation', here are a couple simple ones:
A preferred position and direction in an ordered universe.
Irreducible complexities in living creatures.
Thermal equilibrium throughout the universe.
A universe full of stars and galaxies.
The above is not meant to be an exhaustive list but only an example.
There are things that neither empirical science nor Evolution can explain, and if you don't evaluate the facts with the philosophical assumptions of a Naturalistic World View, they point to a supernatural Creator.
There are also things like the Noachian Flood and the Grand Canyon.
When you look at the evidence related to the Grand Canyon
without Naturalistic bias, It does indeed apear that the Grand Canyon was the natural result of a universal flood that appears to have been partially if not completely supernatural.
If you really are interested in the evidence and arguments excluded by a Naturalistic World View, there are a number of places to find information, including but not limited to
this about the Grand Canyon.
If natural means fail to explain observed phenomena that is believed to be natural, there is either a natural aspect of the phenomena that has not been observed, or the phenomena is supernatural.If all aspects of the phenomena that can be observed, have been observed, and there is still no natural explanation, any further attempt to explain it by natural means is pure conjecture.
If the means for determining whether something is supernatural is to eliminate all possible natural explanations, then it certainly seems that scientists are performing precisely that task. And it seems that those who complain about a "naturalistic" bias should instead be pleased by the persistent existence of that bias. After all, how else is the supernatural going to be located?
The question is, of course, at what point do you call off the investigation and announce "it's supernatural"? That "throw in the towel" declaration doesn't have a very good track record.