Here’s the way I see it. Feel free to disagree.
The first pull, from the floor until the bar is knee height, is controlled, but not neccessarily slow. You simultaneously extend the knees and hips so as to maintain the angle your torso makes with the floor. Your weight should be closer to your heels than your toes and the bar should be as far back as possible, brushing your shins on the way up. Your back should be arched and your chest and shoulders high, with your shoulders in front of the bar. Arms should be fully extended with elbows rotated outwards. In many lifters, maintaining these positions requires the first pull to be fairly slow.
Once the bar passes your knees, you bring your hips forward until the bar hits your thighs (or hips in a snatch). Your back should remain arched, weight still back towards your heels, arms still extended, and shoulders still in front of the bar. During this transition, your hips extend, but your knees will rebend to some degree. This should be done as quickly as possible, since the bar has a tendency to slow down at this point. On the videos, watch the hips and knees very carefully and you should be able to see this happening, just don’t blink or you might miss it.
The second pull follows with the simultaneous explosive extension at hips, knees and ankles and a shrug. Arms should still be fully extended.
Finally your feet should come off the ground to repostion for the recovery, and you pull yourself under the bar with your arms.
Each phase should flow smoothly into the next. Even though the emphasis in each phase is different, they are all just parts of one single motion.