First of all what is lightning?
Lightning is a massive electrostatic discharge between the electrically charged regions within clouds or between a cloud and the surface of a planet.
For lightning strikes that hit the earth, usually highly negatively charged clouds release a lot of negatively charged electrons that flow to the earth.
Normal air is a notoriously bad conductor of electricity, however ionised air is much better conductor. Initially the cloud sends out downward leaders which branch out ionising a path in the air and are kind of "feelers". As seen in the first 16 seconds of the YouTube video.
Once one of these downward leaders reaches the ground, the electrons can flow from that branch into the earth.
Imagine a line of cars at a set of traffic lights that just turned green. Each of the cars is moving in forward but not at the same time and not at the same speed. First the leading car races forward then a few seconds later the next car and so on. The "movement of the cars" starts with the first car and works like a wave back through the other cars. This wave travels in the opposite direction to the motion of the cars.
Similarly with the lightning strike. The electrons are moving from the cloud to the earth (so the "conventional current" is said to be moving from the ground to the cloud). And this current starts at the ground and travels back up a downward leader as seen 16 seconds into the YouTube video.
Lightning rods are generally pointed at the top to help with the formation of upward leaders, which if they meet with a downward leader will close the circuit and cause a lightning strike.
So how does one reconcile the fact that generally speaking people see lightning as going down?
Well the electrons are travelling downward, and the current starts at the bottom and travels upward.
What people see (when they think lightning is travelling down) is probably the initial brightness at the cloud as it begins to send out its downward leaders.
What people see (when they think lightning is travelling up) is probably the current start at the ground and travel back up the connecting downward leader.