I feel some sort of movement restriction or problem on my left shoulder but not sure if that is the source of the problem...
See a qualified sports physician if you are unsure wether both arms and shoulders move in the same way and to the same extent. If your imbalance is not caused by a mere difference in strength (see point 2), but by a medical condition, you may require physiotherapy or some other intervention before you begin or continue with strength training. It is always a good idea to see a doctor before you begin a sport, and it is the common recommendation in all sports. You may be unaware of a condition that can cause injury and possibly permanent damage.
If your arms or legs are different in strength and the stronger limb carries more weight in an exercise (thus preventing the weaker limb to be exercised equally), find a variation of the exercise that you can perform with one arm or leg. In this case, find or create a one-armed pull-down machine, e.g. fix a weight to a rope that you run over the pull-up bar.
Do your exercises with less weight to gain more control over the accurate execution. In the video, you look like you are at the limit of your strength and your body is compensating by moving your arms and shoulders in a way that reduces the strain to the overloaded muscles.
A movement is performed by a group of muscles, some of with are stronger than others when you begin a sport or exercise. For everyday movements, the strong muscles are sufficient to compensate for the weaker muscles in the group, but when you go to the limit, the weak muscles fail before the strong muscles, and this can cause imbalances as in your example and injury. Often the weak muscles are not those that move the weight (such as the bicep) but those that stabilize the movement. Which is why you need to focus specifically on strengthening the stabilizing muscles so you don't build up the weight-carrying muscles beyond your stability. The stabilizing muscles are more difficult to train, because they often need exercises that aren't straightforward such as pull-ups and require specific care in their proper execution. If you begin to train beyond an everyday level, it is a good idea to see a phyisotherapist or training specialist who will help you with developping exercises adapted to your physique.