I'm going to leave early after getting permission from only one of my captain's. Even though I require permission from both.
Why are you leaving early?
The captain told me to do it
It's understandable to think its okay to do something just because someone in an official position does so, but its naive to assume you won't get in trouble for doing so if there are proper procedures set in place. You could end up getting in trouble of fired if someone reports you for doing something in an unofficial manner.
So, I should get permission from both captains. That sounds like the thing to do.
I am glad that you can see how this will protect you. Speak with your other captain as soon as possible about this.