I have a Strong's expanded exhaustive concordance of the Bible and Pisteuo #4100 says - to
have faith (in, upon, or with respect to, a person or thing), to
entrust (especially one's spiritual well being to Christ). It goes on to say, Pisteuo means not just to believe, but also to be
persuaded of; and hence,
to place confidence in, to trust, and signifies, in this sense of the word, reliance upon. In the saving sense of Pisteuo, I agree with that. Saving faith is belief, trust, reliance in Jesus Christ as the ALL-sufficient means of our salvation.
BUT, don't lose sight of the fact that depending on the CONTEXT, Pisteuo can mean just to believe. In James 2:19, we see 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/confidence/reliance upon Christ for salvation. Their trust/confidence/reliance is in Satan, as demonstrated by their rebellion in heaven and continuous evil works. *Keep in mind the Greek for "believe" in James 2:19 is Pisteuo.
No personal surrender to Christ there or a life inspired by such surrender.
In Hebrews 11:1, we read that
faith is the assurance of things hoped for, the conviction of things not seen. Now let's break down "a personal surrender to Him and a life inspired by such surrender." It sounds to me like Vines is mixing the fruit of faith with the essence of faith and "on the surface" that may appear to support salvation by works, which would explain why works-salvationists are very fond of that definition.
When we choose to believe in/have faith in Christ unto salvation (Acts 16:31; Ephesians 2:8) we are entrusting our spiritual well being to Christ, along with placing confidence, trust and reliance upon Him for salvation. That would be a personal surrender to Him.
The life inspired by such surrender which "follows" is the "fruit" of faith. Faith in Christ is the root of salvation and works which follow are the fruit. No fruit at all demonstrates there is no root. As I stated before. If you
believe in/have faith in Christ unto salvation, then you are trusting in Him alone to save you. This belief/faith results in actions appropriate to the belief/faith (to one degree or the other/all genuine believers are fruitful, yet not all are equally fruitful) - but the actions/works are NOT INHERENT in the belief/faith.