What I’ve said is more like “free will is the (indeterministic) causal power of a being to act in accordance with reasons”. Causal powers, their activity, and reasons aren’t defined in terms of free will themselves. Make of that what you will.
Something like that, I suppose. I don’t claim to have this fully worked out. Though I would say that in my view the brain doesn’t have information in the absence of the soul.
It activates its causal power initiating the decided course of action (e.g. if I choose to raise my hand, I activate by causal power of trying to raise my hand; though it was possible for me to decide not to do so). You are correct to say that in my view nothing else causes this to happen (or perhaps more strictly speaking, it is not wholly the product of prior efficient causes). But it isn’t random, because the agent acts for reasons (i.e. with reference to the intrinsic teleology / final causality that motivated the decision).
That would be an instance of the more general correlation between mental states in the soul and physical states in the brain. It does not appear to me to be a metaphysical necessity that mental states are correlated with physical states (for reasons I can point you to elsewhere), but in our case they are via some level of dependence of the soul on the brain.