The word translated believe is found in the Greek lexicon of the Strong's Exhaustive Concordance and is defined as follows: #4100; pisteuō which means "to have faith (in, upon, or with respect to, a person or thing), that is, credit; by implication to entrust (especially one’s spiritual well being to Christ).
The word translated faith is found in the Greek lexicon of the Strong’s Exhaustive Concordance and is defined as follows: #4102; pistis; persuasion, i.e. credence; moral conviction (of religious truth, or the truthfulness of God or a religious teacher), especially reliance upon Christ for salvation; abstractly, constancy in such profession; by extension, the system of religious (Gospel) truth itself:--assurance, belief, believe, faith, fidelity.
The Greek words for "pistis" and "pisteuo" are two forms of the same word. "Pistis" is the noun form, "pisteuo" is the verb form.
The word "believe" can describe mere "mental assent belief," as in James 2:19, or also include "trust and reliance in Christ for salvation," as in Acts 16:31. In James 2:19, the demons believe "mental assent" that "there is one God" but they do not believe/entrust their spiritual well being to Christ; have faith/reliance upon Christ for salvation. Their trust and reliance is in Satan, as demonstrated by their rebellion in heaven and continuous evil works.
Romans 12:3 - For I say, through the grace given to me, to everyone who is among you, not to think of himself more highly than he ought to think, but to think soberly, as God has dealt to each one a measure of faith.