Here are a couple WT comments on the NWT rendering:
*** w91 9/15 p. 10 par. 4 Exercise Faith Based on Truth ***
4 In the New World Translation, the causative form of the Hebrew verb ʼa·man′ is sometimes rendered “exercise faith.” According to the Theological Wordbook of the Old Testament, “at the heart of the meaning of the root is the idea of certainty . . . in contrast with modern concepts of faith as something possible, hopefully true, but not certain.” The same work says: “The derivative ʼāmēn ‘verily’ is carried over into the New Testament in the word amēn which is [the] English word ‘amen.’ Jesus used the word frequently (Mt 5:18, 26, etc.) to stress the certainty of a matter.” The word rendered “faith” in the Christian Greek Scriptures also means belief in something firmly based on fact or truth.
*** w90 12/1 p. 30 Questions From Readers ***
▪ Why does the New World Translation at times render the Greek word pi·steu′o as “believe” (like most translations) and at other times as “exercise [or put] faith in”?
This is done to reflect different shades of meaning that are expressed by the Greek word pi·steu′o.
For example, A Grammar of New Testament Greek, by James Moulton, notes that early Christians clearly recognized “the importance of the difference between mere belief . . . and personal trust.” Both these thoughts can be expressed using the Greek word pi·steu′o.
Often, the different shades of meaning of pi·steu′o must be discerned from the context. At times, though, different grammatical constructions help us to see what the writer had in mind. For example, if pi·steu′o is followed merely by a noun in the dative case, the New World Translation usually renders it simply as “believe”—unless the context indicates something different. (Matthew 21:25, 32; but see Romans 4:3.) If pi·steu′o is followed by the word e·pi′, “on,” it is generally rendered “believe on.” (Matthew 27:42; Acts 16:31) If it is followed by eis, “to,” it is usually translated “exercise faith in.”—John 12:36; 14:1.
This latter rendering (which reminds us that pi·steu′o is related to the Greek word pi′stis, “faith”) is in harmony with a comment in An Introductory Grammar of New Testament Greek, by Paul Kaufman. This work says: “Another construction which is common in the New Testament (especially in John’s Gospel) is πιστεύω [pi·steu′o] with εἰς [eis] and the accusative case . . . The whole construction of εἰς plus the accusative must be translated rather than attempting to translate the preposition εἰς as an isolated word. Faith is thought of as an activity, as something men do, i.e. putting faith into someone.”