Commander, a few things I have noticed about your squat. As you say, yes you need to improve your hip drive, but a few things must happens first. Most importantly you need to increase your hip mobility. Because your hip mobility is not optimal you go into a forward lean causing your knees to go over your toes, thus this lean throws off your angle not allowing you to get below parallel with the heavier weights. All that being said your heels do not come off the ground, which is of course good.
Also, your first motion tends to bend at the knees. The first motion of a squat should be to push your glutes back as far as possible and arch your back hard. Now out of the bottom drive your hips up and through the movement almost to "pop" them through the squat. Finally, you lean forward a bit more on the way up with heavier weights (almost everyone does to an extent and it may be inevitable when maxing), so to counteract this as best as possible as you initiate the hip drive also drive your elbows up under the bar. This will allow you to stay tight along with holding your breath. The longer you can hold your breath the better. I can't see the width of your stance from the way the videos are shot, but a wider stance will most likely help and continue to point your toes slightly outward.
That being said all of the weights seem fairly light for you and you seem to have more strength than what you are showing. I think hip mobility and initiating your glutes back will allow for greater hip drive and driving the elbows up and under the bar will allow you to stay upright and get to parallel with heavier weights. The strength is clearly there for you man. I hope this helps and you don't mind the advice.
Mike Zourdos, Ph.D., CSCS
Florida Atlantic University
USAPL Raw Lifter