I agree that there should be a difference between natural and supernatural in order for those two words to have at least some separate meaning. In order to illustrate my perspective of it, I'll give the Flatland analogy.
As you know better than I, Flatland is the Edwin A. Abbot (1884) tale about two-dimensional creatures (called flatlanders, if I remember right) who live in 'flatland' (i.e., they live in the x-y dimensions) and their experience with three-dimensional objects.
If a sphere were to cut through Flatland, the flatlanders would at first see a small hole, than a larger hole, and then still the largest hole, and then the hole would shrink until it disappeared. From all appearances, they wouldn't be able to explain the widening and then shrinking of the hole other than say a 'hole opened and then closed'. From the flatlanders perspective, the event would be 'supernatural'.
So it is, I think, with any supernatural occurrences in our modern world. We lack a 'four-dimensional geometry', if you will, to describe those features of the natural world which properly explains intelligent design. So, some people come up with concepts such as ID as if it is a supernatural hand at play, when it is really nothing but "a sphere cutting across our Flatland". The design is mixed with randomness much like multitudes of spheres hitting Flatland. There doesn't appear to be any particular and predictable intervention strategy other than here and there, sometimes and sometimes not. Overall, it does appear to be an invisible hand at work, but unlike the work of a loving intelligent Creator, it looks to be somewhat callous and precarious in how, where and when it 'intervenes' in the natural world.
Some people look for patterns of intervention (e.g., Moses who blamed the victim or a previous generation for a previous evil action that brought on the tragedy), but such schemes are fallible. Jesus had gone so far as not to even address why good happens to some while evil happens to many others, other than it does and that faith is required either way.
I think Jesus' words is very enlightening on when and where supernatural events take place. He, of course, didn't deny supernatural events, but just emphasized on living your life in faith and praying to God to intervene. Ultimately, God's will be done.