Comparing the different options
Regularism seems to be a popular view. However, as @T_aquaticus, @gbrooks9 and others have articulated, there are phenomena such as weather patterns, quantum mechanics (including radioactive decay) which are only predictable in aggregate (if at all), yet we don’t consider them to be supernatural. There are also a host of other examples brought up by @PdotdQ and discussed in Predictability Problems in Physics.
I would also add philosopher of science Nancy Cartwright’s work here, where she points out that even in physics, scientific laws are only completely valid in separate regimes - we only have a patchwork of laws connecting different domains, not a complete description of all reality at all levels. We might have to fall back upon the fuzzier notion of “predictable in principle”.
Finally, there also seems to be the problem that any spiritual truths we know entail predictability to some extent. For example, Christians believe that God is consistently faithful, loving, and providential. Thus, to some extent, God’s qualities are predictable and regular. But we don’t regard God as a natural being. I think there should be a way of rigorously distinguishing between God’s predictability and the predictability of nature, but more work needs to be done.
Interactionism is similar to regularism, with an emphasis on the senses. One problem (which @BruceS and @cdods picked up during the discussion) is that in science we rely on instruments which we do not interact directly with. Nobody has ever seen a Higgs Boson with their senses - only aftereffects of its existence. In fact, most modern physics experiments rely on a long chain of logical inferences from the phenomena, through the instrument(s), to the scientist herself.
Finally, there is the problem of miracles (such as the turning of water into wine, or the parting of the Red Sea), which people often regard as supernatural, yet in many cases they are observable by the senses. Are we content to bite the bullet and call these natural (or hypernatural, as @DaleCutler says)?
The scientific-historical definition seems to be popular among agnostics and atheists. If one starts with materialistic assumptions, then this view seems to be fully internally consistent. One critique is that this definition assumes that science will continue to be successful in the future to describe all aspects of reality. But how do we know this? Even in theoretical particle physics, where some are fearing that we are nearing a “particle desert” which will not be surmountable unless we build accelerators which are many orders of magnitude larger than what we have now, if at all. The future success of science seems to depend on an underlying starting assumption that everything in nature is regular and predictable in principle. This is not a philosophical conclusion, but an optimistic assumption.
Secondly, the scientific-historical definition is at odds with how theologians and religious scientists have historically understood the supernatural. Even in the Bible, we see affirmations of regularity and order in the cosmos. These are viewed as attributes of God’s action, not evidence that he doesn’t exist. Thus, it seems that this definition is improperly changing the understanding what the term “supernatural” is meant to refer to. It is a definition that is unrecognizable to anyone who does not already believe in the non-existence in the supernatural.
My own view of personalism hasn’t been discussed much, other than @swamidass pointing out that it could imply that humans are also supernatural. To that I agree, in the sense that our souls are probably supernatural and not fully analyzable by science. I also pointed out that one doesn’t need to hold humans to be fundamentally personal, they could be completely physical (although this might be in conflict with traditional theology).
Finally, @T.j_Runyon’s Creator/Created definition seems to restrict the supernatural to only direct actions of God, the only uncreated being. Everything else is natural. As some others have pointed out, are angels natural? How about the disciples of Jesus performing miracles? One way to wriggle out of this is to argue that in these instances, it is really God, and not angels or humans, who is acting. But presumably many people believe that God can also act through natural means (“Providence”). How do we distinguish natural providence from supernatural acts of God? Or is providence itself supernatural?