Good job!
Biblical faith is not just an act of intellectual assent--that's fides.
Biblical faith is trusting in/counting on--that's fiduces (as in a fiduciary trust).
So are you lay, religious or cleric?
I'm a farmer-and a few other things not all of which should probably be mentioned in polite society. And, I know, Carpenters are better. So...in historical teachings I'm familiar with, the supernatural gifts of faith, hope, and love are defined as
faith meaning intellectual assent, which is why even demons can believe, while
hope means to actually place our trust and confidence in the truths of our faith, in God, and
love is the consummation, the result, the goal, the capstone of it all that truly bonds us to Him.
But that's rather technical stuff. In common usage as well as other teachings I know of, yes, faith means to place our trust in God, perhaps combining faith with hope in that case. It can be taken and clarified even further, as implying
relationship or union with God, the union man was made for and that Adam basically dismissed, not yet possessing an appreciation for it, for Him, in Eden. But that union isn't complete until the next life, when we'll know and love Him fully. It begins here, though, or
should begin here, and our righteousness is fully achieved in any real and meaningful way to the extent that we love Him with our whole heart, soul, mind, and strength- and our neighbor as ourselves. Then our justice or righteousness would happen to be complete, confirmed, as absolute as can be for a created being. That love is what Adam lacked, which would've precluded his disobedience, which would've made disobedience
absurd to him, in fact. Love takes time for us-and all the virtues can and should be grown in, for that matter.
Anyway, while I don't at all think that faith necessarily includes love, which means that it wouldn't, of itself, motivate the works which the new testament treats as being necessary, I can live, a bit uneasily, with an understanding that "true faith" is said to automatically and necessarily include those acts. There are some who understand Sola Fide to mean otherwise, where any obligation to be personally righteous and act accordingly is kept strictly distinct from faith as if one could lack those and still be faithful. As if the sheer act of faith, possibly even only at one time in the past, is sufficient to save a person.