The Cohen memo offered no new revelations about alleged misconduct involving the president, but it shed new light on Mr. Cohen’s role in the various investigations being conducted into Mr. Trump and his inner circle, and on his frequent contact with the president’s legal counsel and advisers as he prepared for congressional testimony that he later admitted was false.

The document provided a detailed, on-the-record glimpse of what took place between Mr. Cohen and Mr. Trump in relation to arranging payments to two women who said during the 2016 campaign that they had previously had affairs with the candidate.

The memo refers to “Woman-1,” who appears to be Karen McDougal, a former Playboy model who was paid by The National Enquirer for her story. The magazine, whose chief executive, David Pecker, is friends with Mr. Trump, then killed the story.

The memo mentions that “Client-1,” as Mr. Trump is described throughout, did not reimburse the corporation that owns the tabloid for the payments to her, as he had agreed to.

The document depicts Mr. Cohen as an unsophisticated and flawed man who was trying to please an exacting, demanding and powerful boss: Mr. Trump. Mr. Cohen’s lawyers wrote that he realized his guilt in lying to Congress about the duration of his involvement with a Trump Tower project he was exploring for Moscow.

Mr. Cohen consulted with White House staff members and Mr. Trump’s “legal counsel” as he prepared for his false congressional testimony, though it did not identify the lawyer, according to the memo. The document did not say if Mr. Cohen was asked to lie, nor did it identify which counsel he spoke with.

But the lawyers wrote that Mr. Cohen had consulted with Mr. Trump’s aides because he knew the president wanted to “dismiss and minimize the merit” of the special counsel’s inquiry, and that Mr. Trump and his aides “were seeking to portray contact with Russian representatives in any form by Client-1, the campaign or the Trump Organization as having effectively terminated before the Iowa caucuses of Feb. 1, 2016.”