I've experienced similar and so have others, imo it was entirely due to how I started it, giving the car a few seconds in the 'ON' position totally solved the 'issue' it's been a long time since I've had a scare, mine would seem to crank and crank and crank before firing. Again, not accessory position, not just key in the ignition, you should be able to hear the fuel system prime before you go to start the car.
I can't tell from the video if you're waiting long enough, if the key is in the 'ON' position before the video starts then it probably isn't the issue, but if the key isn't in 'ON' before the video starts that would be my first guess.
http://www.ft86club.com/forums/showthread.php?t=25822
http://www.ft86club.com/forums/showthread.php?t=102059
http://www.ft86club.com/forums/showthread.php?t=18874
It's habit for me now to put the key in the ignition, turn it to 'on', fasten my seatbelt, take in my surroundings and THEN start the car.
Edit: Definitely more of a problem when the car is 'warm' as opposed to dead cold or hot for some reason.
Lots of people say "No I'm doing it right, Geez" when they're not, no shame in it.