Here are the faults I see.
1. Your upper arm remains horizontal and so there is no flex in the shoulder.
2. You lay your wrist back on the back cast. So the fly rod is horizontal to the ground on the backcast stop, at about 9 AM on this video.
3. Similarly, you stop on the forward cast at almost about 2:30 Pm. So your casting arc is just a little shy of 180 degrees. We call this windshield wiper cast.
4. There is no delay between your forward cast and your backcast to allow the line to unfurl. The line needs to completely unroll on the forward and backcast or else you will not have the opposing mass of the line for the opposite cast. The rod needs to work against the all
the line for a good cast.
You need a delay
between the forward a backward casting strokes. The longer the cast, the longer the delay.
5. Your feet are essentially at the same level. Since you are not an owl, this position makes it impossible for you to turn your head to see your backcast. You can cast like this if you are accomplished. As a beginner, I recommend that you place you right foot back a bit more along a 45 degree line to the baseline.
My teaching philosophy is to teach a basic simple casting stroke called the foundation cast. It is a simple and compact casting motion. Once this is learned, you can add to it.
The basic casting motion illustrated above is very simple. It is a rotation of the shoulder down and the opening (extension) of the elbow joint. It is similar to a karate chop.
Here are illustrations from Jason Borger's Nature of Fly Casting: A Modular Approach
that show the shoulder and elbow rotations that result in a short convex casting motion. First the beginning position of the forward casting stroke.
Then Jason illustrates the positions 1 during a casting demo.
Position 2 is the stop position.
Then Jason illustrates position 2 during a casting demo.
If you compare the casting stroke above to the stop motion photo below will see that this is the exact cast that the angler is making. Notice the curve of the of the rod stroke a the caster's hand goes forward and down. This is because he is rotating his arm at the shoulder and his forearm at the elbow. I cannot emphasize this basic stoke enough.
Practice this stroke. Practice it by using a pen or a pencil to pantomime the motion. Practice it while watching TV. Memorize the instructions below and burn the image into your brain.
The shoulder moves the elbow up and down like a pump handle. Simultaneously, the elbow opens and closes but never opens so much to widen the fly line loop. The wrist does not lay back or forward. Lock your wrist.
Watch this basic cast videos b Mel Krieger. Pay attention to the basic cast he teaches. It is identical to the motions I described above.
The shoulder and not the elbow or wrist is the most power joint in your arm. Cast from the shoulder. The shoulder drives the cast by moving the elbow up and down while the elbow opens and closes.
If you still have a problem with laying the wrist back too far, I can change your rod grip to correct that error.
You can see your forward cast but you must turn your head to see your backcast. This head turn can introduce a curve into your cast, but it is acceptable for you to watch your backcast while you are learning. It will improve your timing.