Here are 3 screenshots during your 3 casts. They are at approximately 4, 5 and 7 seconds, all near the end of the forward cast. Unfortunately you have do not good views of the rod position or the fly line on the backcast.
For beginners, most of the casting problems are on the backcast, so it would have been better to place the camera further away to see the entire cast. The arrows point out casting faults that I will explain later.
I was correct in that you keep your hands together. See the arrow in the first screen shot. That will be a problem when you begin to shoot line. It will tend to wrap around your rod handle or reel. Unless you are double hauling that that motion brings the line hand next to the rod, you need to keep some spacing.
Secondly, there is no need to reach forward at the end of the cast. You do that in an unconscious effort to gain more distance but really, how important can that extra 6 inches be? Stay centered over your center of gravity. If you want to rock, put your right foot back a little more when you rock forward, you are in balance.
Notice in all three photos, there are waves in the fly line and leader. See arrows in photos 2 and 3.
There are multiple things that can cause this. One is starting the backcast either too early or too late. A forward cast should start just before or has as the backcast straightens. Start too early or late and you get waves meaning the fly line was not straight back when you started forward. So you need to work on the timing
There is a variable delay between casts, a longer pause between casts for longer casts. With the right foot back at about 45 degrees, you can turn your head to look at your backcast. Notice in the video that you do not look at your backcast at all.
It is all right to look at your backcast when you are learning, in fact, it will improve your casting.
The second thing that can cause waves in the fly line is the unequal application of power. You are NOT accelerating smoothly and I am pretty sure you are shocking the rod and accelerating much too fast much too early. You need to start slower and end faster. The most rapid acceleration is just before the stop.
I notice that your rod stop occurs when your arms are extended. Stop with the rod higher at about 10:30 on the delivery and then lower the rod to follow the line down. Notice the wide loop
? this is partially caused by the lower stop.
I cannot see the complete backcast in any views. In your first backcast, you seem to stop the rod in about the correct position but on casts 2 and 3, you are breaking your wrist and the rod is flopping back too low. See the open angle formed by the rod butt and your forearm(arrow). That angle should be closed, and the fly rod should be in line with your forearm at the stop. The cure for this is to not break the wrist.
This can be corrected by turning your head to look at the rod position and the track of the fly line on your back cast.
If you cannot help but flop the rod back, I recommend that you change your rod grip to the three point grip.
I wrote an article about the 3 point grip for Wisconsin Trout. See pg. 23 below:
It is explained in depth on these two blogs:
Gary Borger » Blog Archive » Three Point Grip Part I
Gary Borger » Blog Archive » Three Point Grip Part II
So here are my recommendations. Put your right foot back at about 45 degrees so you can turn and watch the backcast. Work on proper timing and proper acceleration. Smooth out the casting motion.
Keep your hands motions independent. They should not move together, but I cannot tell you exactly how to hold them at each phase of the cast without being there with you.
Stop higher on your back cast and forward cast. Do not reach both hands or even you casting hand forward at the end of the cast. Stop higher on the delivery and follow the fly line down after the stop. Do not break your wrist on the backcast.
Here are 2 earlier discussions on casting and fly rods that may be helpful. I suggest you read them and see what works for you. There are a series of videos by Mel Krieger that should help you.
Casting style - action preference...
Over or under lining a rod, what are the advantages and disadvantages?