In what way would Betty's fear of her father have accounted for her behavior after the incident in the woods?
Betty's fear of punishment from either or both her father and Salem society motivates her behavior after the incident in the woods. This becomes clear when specifically referencing her illness. If Betty's illness is seen as legitimate, then her body shutting down is a physical response to the internal state of panic she is experiencing. Parris' own insecurities about his position, title, and wealth have been transferred down to his child, who understands that her actions will have severe repercussions. If her illness is seen as legitimate, then her behavior after what happened in the woods is almost a physical form of her guilt and fear of what will become of her.
If her illness is seen as inauthentic, then it becomes clear that Betty understands the world of trouble she has inherited as a result of her actions in the woods. Betty faking the illness becomes seen as a desperate attempt to avoid the pain and intense rebuke she will experience from both her father and Salem society because of what she did. In trying to misdirect everyone with her illness, Betty's actions become painfully clear that she understood the actions she undertook with the other girls. Similar to the hysteria caused by the naming of names, Betty's behavior after the incident in terms of her illness is an attempt to move blame and shift responsibility from herself to others.