"[For non-reformed theologies] at the end of the day, the security of the believer finally rests with the believer. For those in the opposite camp [Reformed], the security of the believer finally rests with God -- and that, I suggest, rightly taught, draws the believer back to God himself, to trust in God, to a renewed faith that is of a piece with trusting him in the first place."
I'm decidedly non-Reformed (despite recognizing some Reformed as great people of faith in regard to the essentials), but I am not Arminian. I believe, in no uncertain terms, in the absolute security of the believer. The finished work is perfect and absolute and secure.
There may be subtle differences in understanding God's sovereignty, but there is no difference in the understanding of God's ability to keep that which is his.
Might I go a little "ultra" on you for a second? This verse in Ephesians is held by many among the UD as a specific choosing of God in his sovereign choice of "the Body" for his sovereign purposes:
Blessed be the God and Father of our Lord Jesus Christ, who hath blessed us with all spiritual blessings in heavenly places in Christ: 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...
I won't go into "in heavenly places" and "before the foundation of the ages" as we have looked at these elsewhere. Suffice it to say for this context, there are many "non-Reformed" who recognize both man's responsibility to respond to the witness of God (even in the very creation), while also recognizing God's absolute fidelity in keeping that which he has reconciled.
And all things are of God, who hath reconciled us to himself by Jesus Christ, and hath given to us the ministry of reconciliation; to wit, that God was in Christ, reconciling the world unto himself, not imputing their trespasses unto them; and hath committed unto us the word of reconciliation. Now then we are ambassadors for Christ, as though God did beseech you by us: we pray you in Christ’s stead, be ye reconciled to God.
-2 Corinthians 5:18-20
The work of Christ is absolute. God is already reconciled to men through his perfect sacrifice. Men are called to thus be reconciled to God. The offer is real and in accepting that offer, both God and man are reconciled to each other. That cannot be broken. The sacrifice has already taken away all "trespasses" and God is faithful to both his word and his satisfaction in that sacrifice.
I realize there are some semantic differences, but these are very secondary when we dwell on the finality of the cross and resurrection. So, I have no gripe with any Reformed who line up side-by-side to defend the absolute perfection of the sacrifice of the Son of God, who join me in "the ministry of reconciliation," imploring men to be reconciled to God by that work alone.
As a final note, I would add that as Carson worries about the non-Reformed, he has allowed some bad theology to slip past him. He has endorse the gender-neutral Bible translations and was part of the problematic NLT.
Not being Reformed does not make one an Arminian. These are manmade distinctions. Helpful for discussion, but they're not Biblical absolutes from which we must choose. let's be careful with absolutist labels.