I think this is officially called "prevenient grace", though you have not embraced term. You are substituting FAITH for "prevenient grace" IMO. This is 'free willers' doctrine that explains how man can overcome the depravity of man (verses like NO ONE SEEKS GOD).
The testimony of the Spirit instilling faith in the hearts of men so they may trust in Jesus and be saved is certainly an example of prevenient grace.
"6 ...it is the Spirit who testifies, because the Spirit is the truth.
10 Whoever believes in the Son of God accepts this testimony. Whoever does not believe God has made him out to be a liar, because they have not believed the testimony God has given about his Son." - 1 John 5:6,10
Man can't and won't reach out to God. God comes to man through the voice of the Spirit. All it takes is a visitation of the Spirit to enable man to see the truth about the gospel. As we see above, some accept his testimony, while others (most) reject his testimony, in effect calling God a liar. The ability to overcome our depravity and accept the gospel comes through a visitation of the Spirit.
Dictionary definition of FAITH: complete trust or confidence in someone or something.
Similar: Trust, belief, conviction, hope
Thus, I don't except your differentiation of the words FAITH and BELIEF and cite the dictionary as my authority.
Here's the Bible definition of faith:
"1Now faith is the assurance of things hoped for, the conviction of things not seen." - Hebrews 11:1
Through a visitation of the Spirit, an unregenerate person is able to see that which he can't see. Faith is the proof, or evidence that what he can't see is actually true, and so he can make a decision either way as to whether he wants to accept what he can now see, or reject it and go back to his blinded, depraved mind.
I think most Arminians would rephrase your thinking as:
'Believing/trusting' is man's response to God's gracious gift faith "prevenient grace", the supernatural gift of 'knowing'.
No, I would leave it as 'faith'.
Faith (the ability to see that which you can't see really is true) would certainly be an example of God's prevenient grace.
I think I understand you. Your terminology of FAITH is not equal BELIEF is foreign, that's all.
Aside: "prevenient grace" is not found in the Bible IMO, though proponents would site John 1:9, Titus 2:11, John 12:32, Romans 2:4, 1 Tim. 2:4
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I don't have time to look those up right now, but here is God's prevenient grace in the Bible:
"This salvation, which was first announced by the Lord, was confirmed to us by those who heard him. 4God also testified to it by signs, wonders and various miracles, and by gifts of the Holy Spirit distributed according to his will." - Hebrews 2:3-4
Some responded to the testimony of the Spirit, while some (most) did not. God proved to them the reality of the gospel, yet some like the Pharisees turned away in willful unbelief, blaspheming the Spirit (calling him a lying working of miracles) and being eternally condemned as a result. Like all men, because God testified to them, they are without excuse.
Aside2: "prevenient grace" definition: the gift of God to allow men to 'freely' decide to believe or not. (which you are calling faith IMO)
The testimony of signs and wonders of the Spirit is a profound example of prevenient grace--'divine grace given before a man chooses Christ'. This grace is given so men can make an informed decision to accept Jesus as Messiah. The danger being, if a man rejects the evidence and proof of Jesus' Sonship, they are rightfully and justly condemned because they no longer have any excuse to not believe. God is just. It is important that those who he condemns be condemned justly and fairly. That justice does not exist in Calvinism.