Agree with coach... you need to work on quick, quick, quick. We work our catchers with rapid fire blocking so they are drop blocking (leaving the ball where the chest protector blocked it to) then quickly pouncing back into ready position for another rep. We'll hold a broom on top of their helmet during drop so they can feel if they have any/too much lift. This rapid fire drill also helps speed them up with retrieving the ball after blocking. I, too, believe the slow feet contribute to the upward motion before dropping.
When you retrieve the ball, you are bending straight over at the waist instead of attacking the ball, getting next to it so as to be in position to make a strong throw with the correct momentum, bending athletically with the knees and maintaining a balanced, ready-to-throw posture.
It's sort of a bad angle but, like coach said, I also don't see enough difference between your position 1 and position 2 (no runners, runners). With your back side up a bit more in position 2, it will actually make it easier for you to drop without lifting first.
In another video that shows up after this one, a coach is working you at a field with houses in the background. He has you hold in a blocking position and doesn't correct that your back is almost parallel to the ground instead of closer to a 45 degree angle. You want your chest protector to block the ball slightly downward. You don't want to be hunched over so much that your chest protector is practically facing the ground.
Keep up the hard work. I'll be the straight forward jerk here... you are not being pushed hard enough to be quicker and more athletic with all of your movements. You look plenty athletic that I would be demanding more out of you.