There are a few different schools of thought on this topic.
Some say you should drive the knee up and forward at takeoff, keeping the trail leg back to maximize "separation," then initiate the swing with the lead knee as the trail leg comes through, using the knee drive to rotate the hips upward. This works pretty darn well, I used to vault that way. Toby Stevenson is a great example of someone vaulting very high using this technique. This seems to be what is suggested by the article you quoted.
http://www.stabhoch.com/movies/20040827 ... on_590.movHowever, I have recently tried to focus on "locking the heel of the lead foot under the buttocks" at takeoff, rather than letting the foot come forward. What I have found thus far is that this allows a more rigid overall body position at the plant and results in a quicker swing that can stay long all the way up and still get ahead of the timing of the recoil. You need to keep the foot "locked in" all the way to the top of the swing to really feel this effect. The timing changes a bit of the overall vault, but I believe it is better overall. Try it on your short run/soft pole and see what you think. You can also play with this on the rings/highbar and see if it speeds up your swing. The knee still "drives" the hips upward, even when the foot stays locked.
Either way you do it, the key is to not let the lead leg become "an anchor" on your swing...keep the right knee ahead of the trail leg throughout the swing, or it will just be "dead weight" pulling down on your trail leg and slowing down your swing. To directly answer your question, both legs are important in the swing, and the lead knee technically swings first, although the trail leg does the majority of the work. The truth is they should be "locked together" to keep the hips moving upward.
For the following comments: I am assuming the vaulter in the video is you.
Your swing looks pretty good! I wouldn't worry too much about it, because there are other areas you can improve on more.
Your run/carry/drop are pretty good too, right up till your last 2 steps where you stretch your steps dramatically, lose energy, plant late, and slow down. I bet if you drive a little more at the start of your run (longer, more powerful strides), you will be able to finish with quicker, shorter strides instead of the other way around. The timing of your plant will need to quicken up as well to match your increased foot speed...you are not quite fully extended upward at the plant. Other than that you seem to be doing a lot of things well.
Good luck!
Tom