This is an excerpt from a New York Times article.
One prominent family attended the event. Martin Luther King III, the oldest living son of the civil rights leader, and his wife, Arndrea Waters King, met privately with Mr. Biden in Atlanta before the president and the King family, who have planned a series of marches to pressure Democrats to pass the legislation, visited the civil rights leader’s crypt.
During a private meeting with the president, the Kings said in an interview, they did most of the talking.
I think the indefinite article in bold should be changed into 'the' because their private meeting is already mentioned above.
Am I wrong?