I'm struggling to believe that you don't understand that simply creating a category of something, in this case belief, that doesn't follow rules you apply generally to explain things, might not be just explaining by 'magic'.
What category do you think I've created that doesn't follow rules that I apply generally?
As for the second part, the lower level of abstraction cannot work differently from the higher level of abstraction.
New phenomena can arise at greater levels of complexity due to the interactions of the simpler parts - it's called 'emergence'.
Yes, we can talk about free will in a more 'relaxed ' day-to-day' sense...
There doesn't have to be anything 'relaxed' about it, it's a question of getting a logically consistent definition. If we recognise that a part of the (deterministic) universe
is us (let go of the infinitesimal notion of us that externalises everything), then we can say that it is free to do as it wants, which is a perfectly good definition of free will.
...but it cannot make sense to talk about beliefs being caused in exactly the same way and not being choices, which is what the indoctrination approach does.
Beliefs aren't caused in exactly the same way unless you think that the "same way" is the interaction of quantum fields (the most fundamental notion we have at present) but using that 'logic', there is nothing else at all. You've effectively thrown out all the levels of abstraction and emergence in between: particles, forces, atomic structure, chemistry, evolution, biology, physiology, psychology, and the entire human experience - all of which introduce useful and meaningful concepts.
You don't get to say that your beliefs are critically assessed as if that somehow makes them different from all unchosen beliefs without special pleading.
I do my best but I don't recall making that specific claim. I said that indoctrination was a about accepting beliefs uncritically and, for the record, I didn't make the claim that Alan was indoctrinated, I don't know enough about him to know.
That aside, critical thinking has proved to be a useful human tool and there is no justification for dismissing it because everything is (probably) deterministic and all beliefs are caused in the 'same way'. In all the ways that are relevant and important beliefs aren't caused in the same way.