I don't understand the difference between cash out and green up or hedge up

Green up, Hedge up and Cash out (as Betfair call it) all mean the same thing - this is simply where you spread your profit or loss equally across all the selections on that market.

Close trade - is when you put in an opposing bet equal to the amount of your opening bet, this will leave all the profit or loss on just that one selection

For a few examples of these please see this thread on our forum