Appy, you are almost correct in your 2nd sentence. Open theism (as with most everyone anymore) rejects the notion that God "chooses not to know" something.
Along with some other theologians, they've realized that God would have to know what it is that's upcoming so that he can then choose not to know it. Which is illogical.
The point that God doesn't know the future because it isn't knowable is just about right on. I think the open theists would say, "God doesn't know that which isn't true." For example, God doesn't know that I am a helicopter pilot BECAUSE I am not a helicopter pilot.
Therefore, if, for example, Appy has a true free will choice to eat chocolate ice-cream or to eat vanilla ice cream, and Appy has truly not made up his mind or indicated a preference in any way, then God doesn't know what Appy will choose because it hasn't happened yet: that is it is not a fact yet. This is based on Appy's free will choice being truly free will choice.
Another thing to remember is that Appy's choice is entirely separate from God planning certain things in the future and bringing them about by His own power.
(PS: this is all according to Open Theism and not according to me. I'm simply relaying their views as I understand Greg Boyd's writings.)
God knew that Peter would choose to deny him three times.
God has prophesied the future. If God doesn't know than, he is presumption or he forces to future to occur. But God has prophesied man's sin repeatedly, including: Israel's sin, Peter's sin, Sin of man in the last days, sin of man following the 1000 year reign of Christ.
If God prophesied sin, and yet didn't cause it Himself, then God must know the future. If God did cause it Himself, then how can it be sin?
I believe that when God says He is the Alpha and Omega, that He is already the Omega. He exists at all points in time simultaneously. That is why comments are made in scripture like Jesus' "Before Moses I AM" instead of "Befoe Moses I was". That is why future events are often spoken of in past tense in scripture.