If your point is that "grace is manifested in faith in an individual as the undeserved gift" then I don't believe I have made your point for you at all.
For the text of Eph. 2: to agree with your theory, it would say: "For you are saved by grace giving/imparting faith to you." it doesn't say that.
Eph 2:8
For are ye saved in grace (chariti) through faith (dia pisteOs); and that (i.e. being saved in grace by faith)
is not from yourselves (i.e. being saved in grace by faith is not an MO you invented) :
it is the gift of God (i.e. it is the MO God invented and gifted to us).
I could agree with your wording, if you say " In salvation, grace is manifested in faith in an individual as an undeserved gift." But I realise you mean would something quite different by "faith in an individual".
I would say, "Yes, faith (the
ability to trust, that can be used to trust that water will come from the tap when I turn it, or that Jesus died for my sins) is a faculty given to all those born of women, and is therefore a manifestation or gift (charisma) of grace (charis), a gift graciously given by God."
You understand the same sentence to mean that "the belief that 'Jesus died for my sins' is itself a gracious impartation by God to an individual that God wants to save, an impartation He does not give to those He does not choose to save." I don't agree that scripture teaches your understanding.