Ahhh, yes. Where did the "accept Jesus as your Lord and savior," and "ask Jesus into your heart" come from? I hear it mostly from Evangelical Protestants and those from a Fundamentalist background.
19-century revivalism shaped by pragmatism and individualism.
Jonathan Edwards is often credited with the first Protestant with this kind of preaching, but that really compresses something that was more complicated for Puritans. They had broken off from the Church of England, believing the priest's own testimony mattered, because they expected the preaching and example to be sacramental, or at least, to function akin to one. They were intensely religious, scrupulous even. Sin was something omnipresent in their culture, it could be even embodied in peoples and places - they had far more in common with a medieval, enchanted world, than the world of modern America.
Here's one story from Edwards as an example, this one involved a 4 year old girl who had stolen some plums from a neighbors yard, then felt scrupulous afterwards, even though the neighbor gave her permission to eat them:
Edwards describes in A Faithful Narrative, a four-year-old girl named Phebe Bartlet, during the Northampton revival of the 1730s. Phebe had been in a season of genuine distress — retreating to pray alone multiple times a day, weeping, unable to find what she was seeking — when something shifted. She stopped crying and began to smile. She told her mother: "The kingdom of heaven is come to me." She added that three things had come to her, which turned out to be fragments of scripture or creedal confessions : "Thy will be done" and "Enjoy him forever." Then: "I can find God now."
What Edwards valued in the story was not the moment of reception but what it made visible afterward. When Phebe took plums from a neighbor's garden with other children, she refused to eat them even after the neighbor gave permission — and explained, through tears, that she was crying "because it was sin." Another morning her mother asked why she had been crying in the night. "I was thinking about God and Christ, and they loved me," she said. Her mother asked if that was why she cried. "Yes," she said. "It does sometimes." Edwards included this story anticipating disbelief — a four-year-old convert strains credulity in any century — but his argument was phenomenological rather than merely apologetic. The crisis, the barrenness, the sudden gift, the changed comportment toward the world: these were the marks he was looking for. The kingdom had come to her not as doctrine received but as presence encountered, and the encounter left traces.
Puritans framed conversion as God graciously choosing to count you as beloved. It has more in common with medieval mysticism in terms of its intensity and interiority, than the kind of transactional, consumerist logic of "Just accept Jesus as your Lord and Savior", like a proposition, which describes later American pragmatism and individualism. The Puritans believed God worked through means (baptism, preaching, the Supper), but they still expected an inward transformation by grace as the sign of God's calling.
Conversion wasn't a proposition, but something more of a re-direction of the will. And indeed, Jonathan Edwards had alot in common with early medieval theologians in terms of the influence of neo-platonic philosophy - he had studied under the Cambridge Platonists as part of his university education. The Puritans by this time operated by a "half-way covenant", they believed the ordinary means of grace should be effectual if received with faith in producing conversion, but Edwards was concerned that not enough people in his congregation were showing signs of this, so he started focusing more in addressing human affections or felt emotions. In this way he had a very Augustinian focus on human desire, consonant with his Christian neoplatonism, which actually is more important in some ways than his Calvinism.
But the real emphasis with Edvards isn't so much on the conversion experience itself, but what came out of that. The settled faith in God, love of neighbor directed towards the community in practical ways, and so on. The conversion stories he recorded often aren't dissimilar to alot of earlier Catholic accounts of religious conversion.