A video recording released on Saturday of President Trump speaking in 2018 to a group of donors at a private dinner, including two businessmen at the center of the impeachment inquiry, made headlines this weekend for capturing Mr. Trump saying that the ambassador to Ukraine should be removed from her post.

The video was recorded on the phone of one of the businessmen, Igor Fruman. It was made public by a lawyer for Lev Parnas, the other businessman. Both men have worked with the president’s personal lawyer Rudolph W. Giuliani to carry out a pressure campaign on Ukraine, and Mr. Trump has repeatedly claimed not to know them.

The recording undercut those claims, providing fodder for Democrats trying to persuade Republicans to support their calls to expand the impeachment inquiry.

But it also provided a remarkable, inside view at how wealthy and politically connected donors interact behind closed doors with the president of the United States.