Ahhh, that's actually a very good way of thinking about it. Positive lightning should be possible, I just haven't heard of it myself. I've also seen lightning that strikes upward from one cloud to another so the reverse should be equally possible too (one cloud striking another one downward). I'm calling them clouds but I guess they ought to be referred to as bodies of charged air or something else.
As for the issue of the motion of electrons, I personally don't know on that bit. I know that in a wire, there are two types of current. Direct Current is a model in which the electrons are actually being pushed through the wire like it's a water pipe. This is extremely inefficient and lots of heat loss occurs. The reason Westinghouse's DC current lost out to Nikola Tesla's Alternating Current was largely based around efficiency: beyond a certain distance, sending DC power to residental customers is about as efficient as the water company piping hot water to your house.
In an AC circuit the electrons aren't pushed from point to point, rather they effectively vibrate back and forth at 60hz (depends where you live) and do very little traveling despite being energized, so there is less heat\friction. My understanding is that if the power is to be used in a digital device, some form of semiconductor (such as a polarized capacitor) and\or a transformer are needed to groom the current, turning it into DC current, whereas it's used directly as AC current if it's simply powering a large motor such as a vacuum cleaner.
Since a lightning strike can fuse sand into glass, explode a 3-foot-thick tree like a bomb blast, and creates an auditory report heard beyond the horizon line, it doesn't seem improbable to me that the electrons rushing through the air and other substances are responsible for this destructive, fusing power. But maybe it's more similar to AC electricity, and the voltage is just so high that it rapes you anyway?