Salvation belongs to the Lord (Psalm 3:8)

The old testament said the ones with a contrite heart where those who would dwell with God. While the New T says it’s those of faith. Here’s an interesting comparison of both.

Only in Old Testament (Psalms 34:18; Psalms 51:17; Isaiah 57:15); nakheh, "smitten" (Isaiah 66:2). Contrite, "crushed," is only the superlative of "broken"; "a contrite heart" is "a heart broken to pieces." In Holy Scripture, the heart is the seat of all feeling, whether joy or sorrow. A contrite heart is one in which the natural pride and self-sufficiency have been completely humbled by the consciousness of guilt. The theological term "contrition" designates more than is found in these passages. It refers to the grief experienced as a consequence of the revelation of sin made by the preaching of the law (Jeremiah 23:29).

The Augsburg Confession (Article XII) analyzes repentance into two parts: "Contrition and faith," the one the fruit of the preaching of the law, the other of the gospel. While contrition has its degrees, and is not equal in all persons, the promise of forgiveness is not dependent upon the degree of contrition, but solely upon the merit of Christ. It is not simply a precondition of faith, but, as hatred of sin, combined with the purpose, by God's aid, to overcome it, grows with faith.

Definitely the second one you listed is the one that is theologically accurate. The first thing mentioned sounds a bit like the false "works" gospel, which tells people they must earn their salvation, something humans simply cannot do. Yet, when we come to God, repent and believe, and receive that free salvation gift, we begin to change and become more holy because God begins to sanctify us though the Holy Spirit within us. Take care! Thank you for your question.