The concept of free will can be illustrated like this: Imagine there were two parallel universes that were exactly identical, and each contained a person that had exactly alike experiences throughout his life as the other, due to the identicalness of the two universes. Now, looking at one moment in both there lives, for example; suppose they both came to a split in a path that leads left and right. Even though they have both had the exact same experience in life and both are biologically equivalent, free will would say that the outcomes of this moment could be different. That is, one might go left and the other right.
Now, the exact same thing would happen in a demonstration of randomness: two identical universes, two identical people, same situation upon which to decide, but different outcomes.
That they have similar outcomes, free will and randomness, does not necessarily imply they are identical in nature (if you can ascribe a nature to free will). While they share the common characteristic of the future not being set in stone, they differ in that free will contains conscious control, whereas randomness does not.
That, I believe, is the deficiency of both of our demonstrations. They only examine the outcomes, which, for randomness and free will, behave exactly the same way.
The dichotomy you set up, I think, further illustrates the difference between free will and randomness. You said (paraphrased) "If outcomes are not determined, then they is random. Since randomness is not free will, free will cannot exist." The problem here is a contradiction between the first premise and second premise. You posit with the first that there is only determination and randomness, that they are the antonyms of each other, that they are the only two possible causes. But then you mention that free will is not randomness, which implies that free will is a third possible cause. And if free will is a third possible cause, then an outcome might indeed be generated by determination, randomness, or free will, whereas your first premise says there is only two causes possible.