The persons answering "born again," would be correct if they understand the process of being "born again" is 100% the work of the Holy Spirit. John 3:6-8 says, "That which is born of the flesh is flesh; and that which is born of the Spirit is spirit. Marvel not that I said unto thee, Ye must be born again. The wind bloweth where it listeth (the Greek θέλω (thelō), meaning "where it wants to"), and thou hearest the sound thereof, but canst not tell whence it cometh, and whither it goeth: so is every one that is born of the Spirit." Note that our physical birth has nothing to do with our actions or will; so it is with our Spiritual birth.
For those who answer "believer," even the Scripture says that simply believing does not make one a Christian; the devils believe (Jam. 2:19)! However, if their answer is based upon believing the Word of God (Rom. 10:17), which presents the Gospel message that Jesus died for our sins, that He was buried, and that God raised Him from the dead to be Lord of our lives (1 Cor. 15:1-4; Rom. 10:8-10), then they would be right.
As for those who answer a "child of God," here is what the Word says about that:
* John 1:12 "But as many as received (Jesus), to them gave He power to become the sons of God, even to them that believe on His name."
* Romans 8:14 "For as many as are led by the Spirit of God, they are the sons of God."
* Philippians 2:13, 15 "For it is God which worketh in you both to will and to do of His good pleasure. That ye may be blameless and harmless, the sons of God...."
* 1 John 3:1-2 "Behold, what manner of love the Father hath bestowed upon us, that we should be called the sons of God: therefore the world knoweth us not, because it knew Him not. Beloved, now are we the sons of God, and it doth not yet appear what we shall be: but we know that, when He shall appear, we shall be like Him; for we shall see Him as He is."
A genuine relationship with God is 100% grace, 100% His doing, and 100% faith in Christ!