although the beams looks as though it is frozen midway through the bottle, in reality we know it must be much past that point because the light has had time to propagate to our eyes (or the camera in this case) to see it ... the camera is farther away than the bottle is long which means the entire event that we see happened (started and ended) before the video capturing started.
Yes, we see the light only when it hits the eye (via the camera, internet and then your video screen) after scattering from the smoke in the bottle, by which time the main beam will have already passed through the bottle. We wouldn't even see the light if it wasn't for the smoke inside the bottle, otherwise it would pass straight through and would never be deflected into the camera (except where it hits the plastic on entry and exit). In fact we don't even see the light that passes through the bottle at all - we only see the portion of that light which is scattered from the main beam and therefore does not pass through the bottle.
Also, we don't know the real pulse length of the light. I assume it is actually much shorter than the teardrop looking thing that propagates through the bottle in the video. My reasoning is that because we are seeing the pulse seemingly frozen midway through the bottle, the light that we see was scattered to the camera and some of the scattered light must have taken a longer path and hasn't had time to reach the camera yet. The light that takes a longer path will look more spread-out / blurry than the light that took a direct path to the camera. That is why there is a tail-like effect of the beam as it moves. The light at the leftmost part is the light that is just now reaching the camera even though the wavefront of light has moved past that point.
I don't agree entirely with this. Most of the streaky tail/teardrop effect comes from a natural decline (half life) in the intensity of light (i.e., decreasing density of photons leaving the laser) at the end of a laser pulse, as the population of photons bouncing back and forth in the laser "gradually" find their way out of the laser. Remember that the exit of a laser is only partially mirrored, so (for example) 50% of the photons will exit at first pass while the other 50% will bounce back inside the laser, then 50% of that remaining 50% (i.e., 25%) will exit slightly later at second pass, 12.5% at third pass, and so on. This explains the streaky tail.
In fact secondary scattering of light in the smoke will be very rare, which is why the width of the pulse still looks quite sharp. Otherwise if the streaky tail was primarily due to secondary scattering in the smoke, then the blurring of light across the width of the pulse would be as great as that seen along the tail (more like a fuzzy sphere/circle), but that is not what we see in the video.
Therefore, the video shows very well the real shape and length of the pulse.
Still, I think it's incredible that we can effectively watch the pulse of light travel in slow motion at such high resolution (0.3mm per frame!!!), even though the video is "stitched together" by repeating the measurement many times.