Chosen in Him
Being Chosen in Him = God's will for us: "According as he hath chosen us in him before the foundation of the world, that we should be holy and without blame before him in love. Having predestinated us unto the adoption of children by Jesus Christ to himself, according to the good pleasure of his will."
God chose us, but it is our free will to choose Him.
Our Free Will
Having Free Will = Either Doing His will or not doing His will: If we are doing His will then we should be holy and without blame before Him in love. If we are not doing His will, then we are not being holy, are not without blame before Him in love.
One can reasonably conclude that it is when we choose Him and do His will that our names are written in the book of life, and that the phrase "book of life from the foundation of the world" refers to the book itself existing before creation, and that it is part of the overall plan of salvation, which includes "...the Lamb slain from the foundation of the world" and "According as he hath chosen us in him before the foundation of the world, that we should be holy and without blame before him in love."
But what I don't see in your explanation is an exegetical argument from Scripture, or a rebuttal of what I have already offered to the contrary. Respectfully, the ability to describe a point of view is no indication on its own that it is actually what the biblical authors
intended to communicate. The latter is what must concern us.
The structure of Paul's argument in Eph. 1 simply does not allow for human choice to
precede or
condition divine election. Notice the order of clauses:
καθὼς ἐξελέξατο ἡμᾶς ἐν αὐτῷ πρὸ καταβολῆς κόσμου, ἵνα ἦμεν ἅγιοι καὶ ἄμωμοι...
God
chose (ἐξελέξατο) before the foundation of the world
in order that (ἵνα) we might be holy and blameless. Holiness isn't the
basis of being chosen; it is the
intended outcome. Likewise, verse 5 grounds predestination "according to the good pleasure of His will" (κατὰ τὴν εὐδοκίαν τοῦ θελήματος αὐτοῦ), not ours.
Human faith and obedience are therefore
the fruit of election, not its cause. To invert that order is to ignore what Paul wrote.
As for the Book of Life, Scripture never describes anyone's name being newly written
because of faith; it speaks of those whose names "were written from the foundation of the world" (Rev. 13:8; 17:8). The "foundation of the world" modifier applies to the
writing, not the
book. That means God's saving decree precedes both creation and human response.
So yes, believers truly choose Christ and obey Him. But they do so because God first chose them in Christ. Election isn't God's response to human decision, nor does it erase human decision; rather, it is the divine cause behind it.