The Greek words for "pistis" and "pisteuo" are two forms of the same word. "Pistis" is the noun form, "pisteuo" is the verb form. If you believe in Christ (Acts 10:43; 13:39; 16:31); have faith in Christ (Acts 26:18; Romans 3:22-28; Ephesians 2:8) then you are not only believing in the historical facts about Christ and the totality of His person, but are trusting in Him as the all sufficient means of your salvation.
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 word translated believe is from the greek word 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 "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, nobody is questioning the fact that 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. Saving belief/faith is more than just an "intellectual acknowledgment" to the existence and historical facts about Christ. Saving belief/faith trusts exclusively in Jesus Christ as the all sufficient means of our salvation.