I often see people doing pull-ups / chin-ups with knees bent such that their legs are "hanging" behind. Shouldn't the legs be straight down, perpendicular to the ground?
Where should our legs be when we do (standard) pull-ups?
Where should our legs be when we do (standard) chin-ups?
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.
Adding this answer just in case another person stumbles across it.
If you're doing weighted pullups, it becomes pretty important to keep your back straight. Your body tends to whipsaw around a bit when you're doing pullups fast, and the added weight dangling off your waist can pull and yank on your back in un-fun ways.
If you keep the weight dangling from a chain, I find it pretty handy to keep the weight close to your body to minimize the whipsaw effect. This kind of distance has the thing flying all over the place:
Versus this one, where the weight is:
Depending on your goals, once you can do ~10 regular pullups, consider adding some weight. I'm always scratching my head and people who do all kinds of cable and machine work and leave out weighted pullups (and dips). They are phenomenal compound exercises.
