Yes! I like that video a lot!
What is funny is the person who blogged about the video is in rehab from hip arthroscopy for a labral tear (which I've had done on both hips). So that is one of the things I have been trying to improve on the most.
But yes -- that's a good example of Trendelenburg. To simplify it, what's happening is your hip abductors (particularly the glute medius) are not functioning at the level they should be for whatever reason (trigger points, intra-articular instability within the hip itself, lumbar/sacral alignment issues, etc).
When you stand on one leg, a healthy hip should be able to keep that level plane in your pelvis so your center of gravity remains in alignment with your spine. However, with a hip that has weak abductors, it cannot maintain that stability so the opposite hip (in the leg being picked up) dips downward which creates instability.
Even though this instability might not be extreme during running, even a slight weakness in itself can pose for major problems in your lower extremities because you are obviously taking thousands of steps for each run you go on. If the forces transmitted through your leg are not efficient, it's going to cause injuries down the road.
I haven't heard of the gait guys I am definitely going to check out that second website you posted, thank you!!