I could be wrong but those cognitive processes you refer to might be "elaboration likelihood model" basically:
Central processing is when the person thinks deeper about something.
Peripheral or side processing is when someone thinks quick and makes a decision based on how attractive the situation is presented, reframing the question as you said.
I just quoted the article, but it makes a lot of sense to me that people probably use "willpower" to refer to a few fundamentally different things, which is why we still have neither a solid theory of "willpower" nor ways to train it reliably.
There is probably the component of "does this person react immediately, or do they slow down and consider things first?" Which may be partially a habit, and partially biological.
Then there is a question of the mental model the person has. If one person believes that "X means Y",... (read more)