It's good that people are asking this question about a connection between terrestrial lightning and space. There are many good reasons to believe this is happening. This one in particular stands out:
"Lightning in clouds, only a few miles above the ground, clears a safe zone in the radiation belts thousands of miles above the Earth"
The tendency, however, is to point to cosmic rays originating from stellar explosions as the source -- as if this is the only electrical activity happening in space. What is skipped over in these articles is the more fundamental inference that Earth simply exhibits a net charge, and an electric field as a result of that charge. An E-field is the result of a buildup of electrical charge. As you know, like charges tend to repel one another, so a buildup leads to an electrical force between that charge and the ambient charge associated with the surrounding space.
That would seem to be yet another worthy explanation, in part because it could also explain why the Van Allen radiation belts are there at all. The existence of "belts" of differing types of charges, side-by-side with one another, which never apparently recombine, is best explained with the laboratory plasma physics concept of double layers. And one of the features of double layers is that, on an astrophysical scale, they have an ability to accelerate charged particles. So, if you are seeing what appear to be double layers around Earth, this also raises the question of what is accelerating the cosmic rays.
But, notice that double layers remain an obscure topic in astrophysics, and in science journalism -- even though they represent a core concept in plasma physics. And there exists an unfortunate contradiction here: Despite the fact that scientists admit that 99%+ of the matter they observe in space is matter within the plasma state, and despite the fact that we've known this since the 50's when we first sent probes into space, what has yet to happen within the community of science journalists is to explain to the larger public what plasmas in laboratories tend to do. This has led to extraordinary confusion on the topic of electricity in space.
A person need only carefully watch any of the higher resolution YouTube videos which show the behavior of novelty plasma globes to understand why so many people are talking about plasmas today:
Look closely at those videos. Notice that filaments naturally form from nothing more than the presence of charged particles within a gas, subjected to a voltage potential. That, in itself, is extraordinary. What keeps those filaments together? Like the Van Allen radiation belts, there are no physical wires to contain this charge into a structure. We need plasma physics concepts like double layers to explain these structures -- including the bolt of lightning itself.
But, the story becomes even more perplexing when a person looks even more carefully at those videos, for when the filament touches the glass, it becomes apparent that this is really two or more separate filaments which are wrapped around each other, tight as a rope. So, not only is there a long-range attraction between these filaments -- but there also exists a short-range repulsion which prevents the filaments from combining. How does all of this structure arise without any condensed matter to guide it? It's antithetical to our expectations for the behavior of gases, but there it is. It happens. All of this structure essentially forms itself, quite naturally.
Scientists may very well be able to formulate a theory which explains lightning with cosmic rays from stellar explosions. But, when scientists propose their inferences for lightning, they should also be looking to the phenomena which we observe in plasma laboratories, and even in novelty plasma globes, when we are studying Earth's electrical activity.
For more information, see https://www.youtube.com/watch?v=NenomdAZ11M .