Student or Learner
I learned "believe in" means "to accept the existence of something", but seem to have seen some other cases similiar to mere the meaning of "believe people's words or actions". Is it always strictly "believe the existence of something"?
ex1)...If you believe in your fitness potential, you are likley to live up to your own expectations.
ex2) Do you believe in God?
"Believe in", as you have used it in both of your examples, is a PHRASAL VERB.
"Believe in" can mean any of the following:
1. have faith in the truth or existence of : I believe in God.
2. be of the opinion that (something) is right, proper, or desirable : I don't believe in censorship of the arts | he didn't believe in sex before marriage.
3. have confidence in (a person or a course of action) : he had finally begun to believe in her.