John 1:12 is attached with "a personal surrender to Him" in Vine's definition. John 1:12 - But as many as received Him, to them He gave the right to become children of God, to those who believe in His name. There is the personal surrender to Him. 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 in Him for salvation. That would be a personal surrender to Him, resulting in justification (Romans 5:1) and that is the ROOT of salvation.
The FRUIT of salvation would be living by or "out of" faith, as we see 2 Corinthians 5:7 attached with "a conduct inspired by such surrender" in Vine's definition, which reads - For we walk/live by faith and not by sight. This is our continued walk with the Lord by which by or "out of" faith believers bear fruit/produce works. So faith in Christ is the ROOT of salvation and "a conduct inspired by surrender" which FOLLOWS and produces good works would be the FRUIT. Your understanding of Vine's results in mixing Cause and Effect.
Your definition of pisteuo amounts to salvation by grace through faith PLUS WORKS, contrary to salvation through faith, NOT WORKS (Romans 4:5-6; 11:6; Ephesians 2:8,9; Titus 3:5; 2 Timothy 1:9 etc..).
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. *Nothing there about a life inspired by such surrender which follows saving faith in Christ/fruit of faith/good works being mixed in with the definition of Pisteuo.* In the saving sense of Pisteuo, I agree with Strong's. Saving faith is belief, trust, reliance in Jesus Christ as the ALL-sufficient means of our salvation.