We cannot choose to believe. Belief is a state of mind. You can have a hypothetical state of mind, but only one actual state of mind.
Ostensibly agreed. That is precise why they are a comparison of apples to oranges: one is an attitude of truth towards the proposition--a statement with potential truth value--and the other is an imagined proposition, incapable of being the bearer of truth value.* Their common denominator may be the loosely termed "mental entities" (states of mind treated as objects for discussion), but not belief, which would be one of the mental entities.
Our state of mind is a consequence of not only our actions but also others actions, our preceptions, and our previous states of mind. We can not choose all of these factors therefore we cannot choose our beliefs.
The terminology and the epistemic modelling has to be clear for this discussion to work.
Once the world (the propositions, perceptions, actions, etc.) has been concretized (with the appearance of truth) and become apparent reality, there is no option that will
satisfy to make any part of reality a choice: yes. But part of this discussion touched upon when the apparent reality has actually changed, such as when we realize that what we believe is mistaken. One person's vernacular might capture that as that our belief has changed, while another might capture it as that the information changed and belief remains unchanged.
I will continue later, I have to get to work.
*The discussion of whether imagined propositions are capable of being the bearer of truth value is dealt with as
philosophical realism.