James 2:19 states:
“You believe that there is one God. Good! Even the demons believe that – and shudder.”
~ In Koine the word used for believe is pisteuō (πιστεύω). It can also mean: to trust, to have faith in.
19 σὺ πιστεύεις ὅτι εἷς ἐστιν ὁ θεός, καλῶς ποιεῖς· καὶ τὰ δαιμόνια πιστεύουσιν καὶ φρίσσουσιν. 
~ Note however that in James 2:19, that the word pisteuō is not translated as trust. It does not say or associate trust with demons. What we can get out of this is that the demons believe in there being one God, but they do not trust that there is one God. Something more is implied in saying that trust is involved. This means that the word trust does not mean exactly the same thing is believe.
~ Now lets go to one other verse, Romans 4:5 in English and in Koine:
“However, to the one who does not work but trusts God who justifies the ungodly, their faith is credited as righteousness.
5 τῷ δὲ μὴ ἐργαζομένῳ πιστεύοντι δὲ ἐπὶ τὸν δικαιοῦντα τὸν ἀσεβῆ λογίζεται ἡ πίστις αὐτοῦ εἰς δικαιοσύνην· 
~ Here pisteuō is translated as trusts and faith.
So all I am trying to say is this. In Koine, though the word often used in a given verse is pisteuō, in English, it may be translated as faith, believe, trust, etc. Perhaps this should not be surprising. Why? Well, we do this in English as well with some words. For example, the word you can have a singular referent or a plural referent. Or the word love, can refer to something romantic or to friendships or to a fathers love for his child. And so on… We simply understand from the context what is going on. And …
It seems to me that the Greeks were like that with the word, pisteuō. They simply understood from the context what was going on.
 Nestle, E., Nestle, E., Aland, B., Aland, K., Karavidopoulos, J., Martini, C. M., & Metzger, B. M. (1993). The Greek New Testament (27th ed., Jas 2:19). Stuttgart: Deutsche Bibelgesellschaft.
 Nestle, E., Nestle, E., Aland, B., Aland, K., Karavidopoulos, J., Martini, C. M., & Metzger, B. M. (1993). The Greek New Testament (27th ed., Ro 4:5). Stuttgart: Deutsche Bibelgesellschaft.