What prompts Patria to become involved in the revolution? How does her commitment differ from Minerva's and María Teresa's?

Patria became involved in the revolution as a direct result of something she had witnessed first-hand. While on a mission trip (Patria is the most religious of the girls), the group attending the retreat is bombed by Trujillo's men. Seeing an innocent man die for no reason changed her life, her perspective, and made her doubt her faith in God (temporarily). As a result, she came to understand that she couldn't just sit back and be a spectator. Against his better judgement, her husband Pedrito becomes involved as well.

Sorry, as for the second part of your question, Patria becomes involved in the revolution because of events rather than beliefs. She is very religious, her family is the most important thing in the world to her. Thus, Trujillo's actions didn't prod her into becoming a part of the revolution until it began to affect her personally. Her sisters were involved in a cause because they had been recruited and believed in what the cause stood for. Patria joined the cause because of an individual reaction to the atrocities she both witnessed and experienced.