It doesn't really matter much - you are lifting the same weight no matter how much (or little) you shift your center of mass by moving your legs. Excessively rounding your lower back might have other consequences, though I haven't found any references to back this up. However, keeping your legs straight will activate your abs (especially upper abs in my experience) more, which means a better workout. In fact, this article compares abs muscle activation for various exercises using EMG. In shows, somewhat surprisingly, that a bodyweight chinup activated the rectus abdominis (main "abs" muscle) the most out of all exercises. It also proves that hanging leg raises (hang from bar and raise your legs to 90 degrees) are a good abdominal exercise, too. So, for the maximum workout you could try to keep your legs at a 90 degree angle (ie. parallel to the ground) while doing pullups (I've seen this referred to as an "L-seat pullup"). This is a good, almost full-body workout, but I wouldn't do it if you are unprepared. Be careful to maintain proper tension throughout the movement else you risk injury.
In short, a strict-form pullup is one where your body is in a straight line throughout the movement. [the fact that everyone does them in a certain way does not mean they are right; eg. in the link you gave, the guy in the video never actually gets his chin over the bar, he just touches it at best]
Of course, if your bar is not high enough you will have to bend your legs.