“Believing” Meant Trusting in Someone’s Honest
The verb translated as “to believe” is pisteuo (πιστεοῦ). This is from the same root as the noun, pistis, discussed above. Jesus uses this verb in six-nine verses. This means that he uses it more than the entire Greek OT, in which it appears only forty-eight times. Again, this is evidence that this word was understood to have little to do with religion as such. As with pistis (see this article of the word translated as "faith"), pisteuo is much more popular in the New Testament where it is used two-hundred and forty-one times. Notice that this is exactly the same number as the noun form, which is an interesting coincidence (if you believe in coincidences).
Pisteuo means "to trust, put faith in, or rely on a person," "to believe in someone's words," "to comply," "to feel confident in a thing," and "to entrust in a thing." If we look in Strong’s, we again see that its primary meaning “to think to be true; to be persuaded of; to credit, place confidence in.” It only came to mean religious belief after the New Testament, a long time after Jesus and those writing the Epistles.
As we say with the noun form, pistis, Jesus usually used uses the verb in the context of trusting someone and especially the honesty of their words. It is interesting that when Jesus uses a negative with this verb, he usually uses the objective negative, the negation of facts. Since we are talking about an opinion here, you would expect him to use the negative of opinion, which is a different word in Greek (see this article). So he saw this trust as something more real and solid than mere opinion.