Please note that I only said we can “choose what convinces us” if “convince” is defined in a certain way, and I only noted that because your question to me presupposed (falsely, in my view) that there is something that “convinces” us for every belief that we have, determining us to believe it. I don’t think that’s the best way to phrase the issue. So, instead, I’m going to answer as if you’ve asked me for an example where we can choose what to believe.
Here’s a self-referential example, for kicks. Suppose you have been reading the philosophical literature on doxastic voluntarism, because you want to decide whether involuntarism or some form of limited voluntarism is true. (We both agree that unlimited voluntarism is obviously false, so there’s no need to keep that on the table.) And the reason you want to determine which of these is true is because there is a moral decision you have to make which rests on the answer (deciding whether someone is culpable for some action which they performed because of a false belief they held, say). Now, in your reading, you’ve made yourself aware of the arguments for and against, and you think there are some good arguments on both sides, so that you are not compelled by the arguments in either direction.
My claim is that, in this situation, you are able to choose to believe the arguments for or against the position - not just to make a decision as if one side or the other is true, but to actually come to believe one side over the other, even if you believe it without complete certainty. I base this claim on introspection of my own experience of deliberating about beliefs. And this is pretty much analogous to situations of deliberation over a decision more generally - i.e. cases where I believe we exercise our free will.
Correct. I characterize libertarian free will as the power in some circumstances to act as an uncaused “first cause” of a series of events (uncaused in the sense that one was not caused to act). But the even though I am not caused to act, the act of will is not random, because it has intrinsic teleology - it was done for a purpose; it has a “final cause” in Aristotelian terms.