The best translation of Pisteuo is trust. Trusting Christ makes the whole matter dependent upon Him. Cleansed by Him, full of Him, empowered by Him. Slaves of the Master.
"But now that you have been set free from sin and become slaves to God, the fruit you get leads to sanctification, and it's end eternal life." Romans 6:22
The truth of Pisteuo is freeing.
So your saying you don't agree with the strongs and the Vines definitions.
Because you said Pisteuo is "trust", and trust is a noun which would be the Greek word Pistis.
trusting or trusting unto Christ is pisteuo. But we are providing the act of Trusting, not trusting in His trusting like your kind of saying.
The best picture of "pisteuo" is an old testament word for Faith. "the leaning on a staff with all of our weight behind it. That means our life is part of the act being of trusting. That is why the Definition of "pisteuo" by the vines is so important. Vines:"A personal surrender to Him". Surrender of what? Our lives!
You mentioned we are "slaves of the Master", we'll have you ever known any slaves that stand around watching the master do all the work. Yes, Christ has done a work that is the Grace deposit, "the staff". we are "faithing" into Him by leaning our entire being on the staff which is Christ. The leaning on that staff is a continual action "WE DO" which is what Pisteuo means and is defined in the strongs and the vines.