I don't have time to read every single post today but I did find this thread very entertaining so I did read a lot of them. I noticed that you thought a partial squat maybe 1/4 of the way down with a weight that you can just about manage to hold across your back was a good exercise for improving your ability to jump high.
OK some advice - first of all you want power not strength so lift a lighter weight and lift it faster because this actually works towards what you want - a higher jump. Secondly why the squat when you said it yourself that you are trying to neglect your upper body to reduce your bodyweight? Your upper body is working quite hard to hold that weight in place during the squat.
You can work on your legs without building your upper body by picking different exercises and theres a lot to choose from. Thirdly it is not true to say that a quarter squat is harder than a full squat. I can lift way way more on a quarter squat than a full squat (as much as 100 pounds more) without even training it in a specific manner very often and I think you will find the same goes for everyone else who tried to compare the two.
Finally your squatting technique looks pretty bad, even when you lower the weight 1/4 of the way down, your knees are coming right past your toes which is too much too soon in the movement. If you do take the advice to squat lower that some posters have given you then you are really going to need to sort out that form but I am guessing you will not take that advice firstly because you believe your way is better and secondly because of your injuries.
So whats wrong with doing some leg press with a bigger range of movement with lighter weight with explosive power? You can train your core seperately if you feel the need to although, as you say, you don't have much bodyweight restricing your movement.
Sometimes the best sports specific training is to train as close to what you are trying to get better at as possible. Perhaps try jumping for a dunk with a weighted vest on?
Anyway I hope you like the advice and information which I saw fit to give you. Notice that I have not been mean to you so no need to crack up at me like with some other people who may or may not have deserved it previous to me. Please do keep us updated and I will endeavour to read the rest of this thread later.