Still don't understand what supernatural means

Some time ago I wrote that I don't really understand what supernatural is supposed to mean. Recently I took the opportunity of a discussion on whether humanism should or should not include a commitment to naturalism to pitch the question of what the opposite - supernaturalism - would entail. Although a few answers were helpfully provided, I sadly find none of them very useful.

Basically, I have the suspicion that although the word is continually used in rather important claims, such as that science has nothing to say about the supernatural, or that, see above discussion, one should not carelessly dismiss the supernatural, nobody actually has a clear definition of the concept supernatural.

Of course, a lot of people have examples that pop into their heads when they hear that word, and they would be tempted to reply with "something like demons and telekinesis". But that is not actually a definition, and my question is one step further back. Why would it be considered to be useful to call demons and telekinesis supernatural as opposed to natural? What is the difference between demons and horses, or between telekinesis and gravity, that would make it meaningful to call the first in these pairs supernatural and the second natural and, and this is where the important consequences come in, justify the conclusion that they are consequently beyond the scope of science?

To see what I am getting at, imagine that there is a thing or process called Ulkjam. You have never heard of it before, but I know all about it except whether it should be classified as natural or supernatural; that is what I want to know from you. You can ask me about all of its characteristics and attributes, and I will be able to answer. So what would your question(s) be to help you figure out as quickly and decisively as possible whether Ulkjam is supernatural or not?

So far, nothing useful has come up:

Supernatural is the violation of natural law

Massimo Pigliucci implied, without saying it directly, that supernatural events are those that violate natural law. There are, however, several problems here. Perhaps the most obvious one is that that is merely pushing the question one step back. If Ulkjam follows some laws (shorthand for exhibiting some regularity and predictability, which might even be randomness, as is the case with many perfectly natural processes), why are those laws not also natural laws? So we do not really gain anything by that answer.

(If Ulkjam shows no predictable behaviour at all, then nobody can know or say anyting about it anyway and the issue never arises, see "non-existence" section below.)

Worse, it has happened quite often throughout the history of science that something violated known natural laws. When that happens, scientist do not conclude that something supernatural has taken place. Instead, they conclude that they must have gotten the natural laws wrong and start revising them.

So there are at least two reasons why this definition of supernatural won't work.

Supernatural is when we don't understand the process

This is essentially the same as the previous one only in slightly different words. The claim is that demons for example would be supernatural because they wouldn't work like known biological organisms, and reincarnation would be supernatural because there is no known process by which it would work.

The same objections apply. First, demons and reincarnation, if they existed, would function through some process, and that process could be studied (if they existed), and then it would be understood. It is not understood only because they actually don't exist and thus cannot be studied. The question why reincarnation, if studied and understood, would be supernatural as opposed to one more natural process like digestion or reproduction, or why science would have nothing to do with it if it existed, is left unanswered.

Second, all processes we consider as natural were once not understood. Surely nobody claims that fire is supernatural just because our ancestors a thousand years ago didn't know what happened chemically when wood was consumed.

Supernatural things are things done by supernatural beings

One commenter said that if god does something then it is supernatural. This is begging the question because it doesn't explain what is so different about gods compared to all other intelligences that they aren't part of nature, and, remember, cannot be studied by science while all other intelligences can and are. Onwards.

Supernatural is things that don't exist

This was perhaps the oddest answer of all. Of course the commenter did not phrase it quite like that, but in practice they mentioned things for which there is no evidence that they exist and then claimed that that is why they are supernatural. Problem is, there were many ideas about natural things that might have existed but then turned out not to: luminiferous aether, phlogiston, Bigfoot, etc. That doesn't make them supernatural, so obviously there must be more to the supernatural than lack of evidence if it is supposed to be a meaningful concept.

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Ultimately, it always seems to boil down to this: supernatural is a label that is arbitrarily attached to some things to allow people to make wild claims about them.

Religious apologists (and, sadly, some philosophers) tack the label onto nonsensical beliefs to mark them as beyond reason: this is supernatural, so science no go here. Go away, you are not supposed to say that there is no evidence. Evidence doesn't apply by definition because it is supernatural. But why is it supernatural? Well, because the religious apologist says so.

That is not what one would call reasonable, so it is perhaps impolite to claim that that is what happens. But until somebody can come up with a definition of supernatural that is not (1) obviously insufficient because it also applies to natural things or (2) begging the question, what else am I supposed to conclude?