The way I was taught, an independant seat means that you can maintain your seat independantly of what your legs or arms might be doing - I.e. You don't grip the reins if you are off balance, etc.
We get kids doing work without stirrups and without reins, to establish this. Trotting/cantering over ground poles and then small jumps with your arms out to the sides (Need a steady and trustworthy horse!) and then without stirrups, etc.
I second this... and to add to it; You need to do LOTS AND LOT AND LOTS
of riding. I will admit it, when I was a kid I was all over the place, the best progress I made hands down was when I was riding under a trainer. I was trail riding with her for a good 6-7 hours a day everyday. When we had 2 year olds that needed miles she would ride them and follow my horse at the walk and jog (he was steady and broke).
The older horses we loped all over god's creation.
And we rode everywhere, and I mean everywhere... up and down hills, over logs, through brush, we jogged through town, past the quarry, past the shooting range, over to the neighbors to ride with them, through streams larger rivers and we swam in a pond, we went over and under bridges, along major roads, we rode past every game animal in Eastern PA and then some, we went through ditches, rode in an indoor, along cow pastures... all of this made me a much stronger rider.