Why does the eclipse in this video look annular?
It is quite clear from the film (and the awful soundtrack) that a total eclipse was observed. The Sun's corona and prominences cannot be seen during an annular eclipse. Nor is there a sudden fading or drastic brightening at the beginning and end of an annular eclipse.

The accompanying article talks about "totality" and removing safety glasses to look at the Sun during totality, and that the period of totality was 1 min 53 seconds; the (frankly, annoying) narrator shouts the word "totality" about 23 times.

The eclipse was observed towards the end period of totality (what I mean by that is the shadow is tracking west to east across the Earth's surface. That means the Sun was above the western horizon as seen by the plane, and at 5:35pm, would be 10 degrees above the horizon. That means that in order to observe this event the plane must have been closer to the moon than a point on the surface of the Earth where totality would have been observed at the same time. That means the period of totality would have been slightly longer and the moon would have had a slightly larger angular size as seen from the plane. 

If I had the exact latitude and longitude I could tell you just how much longer the eclipse was. Messing about with Stellarium, I can get a reasonable match of the eclipse time at W157 58 35, N31 22 00. At 10,000m the eclipse is total and lasts about 70s. At sealevel at the same latitude it lasts maybe a few seconds longer (it's difficult to be sure). But the sea level region of totality observed at the same time would have been somewhat further east (but still along the path where totality was observed), where the period of totality would have been shorter.