That attempt, Mr. Cohen says on the recording, came months before he brought Mr. Falwell “to the table” for Mr. Trump. Until then, he adds, “none of the evangelicals wanted to support Trump.”

There is no evidence that Mr. Falwell’s endorsement was part of a quid pro quo arranged by Mr. Cohen. Indeed, the relationship, if any, between the endorsement and the photo episode remains unclear. But the new details — some of which have been reported by news outlets including BuzzFeed and Reuters over the last year — show how deeply Mr. Falwell was enmeshed in Mr. Cohen’s and Mr. Trump’s world.

And they add another layer to one of the enduring curiosities of the Trump era: the support the president has received from evangelical Christians, who have traditionally demanded that their political leaders exhibit “family values” and moral “character.” Mr. Falwell’s father forged those words into weapons against the Democrats after he co-founded the Moral Majority political movement, which propelled Ronald Reagan into the White House and made religious conservatives a vital constituency for any Republican who would be president.

By the time Republicans cast their first votes in 2016, Mr. Trump was starting to show surprising strength among some white evangelicals. But with Mr. Falwell serving as the torchbearer of his father’s legacy, his endorsement became a permission slip for deeply religious conservatives who were attracted by Mr. Trump’s promises to make America great again but wary of his well-known history of infidelity, his previous support of abortion rights and his admission that he had never asked for God’s forgiveness.

“For those of the more fundamentalist variant of evangelicalism, the Falwell family, and the Falwell endorsements, are an important factor,” said Jim Guth, a political-science professor at Furman University who has long studied evangelical politics. Mr. Cruz still managed to win in Iowa. But Mr. Trump soon won South Carolina with strong evangelical support, sending him on a solid path toward the nomination.

Three years later, Mr. Falwell remains an unwavering Trump supporter. Last month he went so far as to suggest that the president deserved an extended term as “reparations” for time lost to the Mueller investigation. In turn, he has had entree to the White House, providing input on education policies that stand to benefit Liberty.