The quoted remarks referencing the Pulitzer Prize are from the description given by the poster (LBJ Library) of the YouTube interview with Woodward. I personally don't think that the distinction being drawn in the Pulitzer award is very important; but since you raised it, here's the information that I found.
The 1973 Pulitzer for Public Service was given to the Washington Post based on the cited "Winning Work" of Carl Bernstein and Bob Woodward. The Pulitzer Prize Board website FAQ states that "The Public Service Prize is always awarded to a news organization, not an individual, although an individual may be named in the citation."
So, the publisher received the award
for the cited work of the journalists in 1973. The Washington Post considers that award to be shared by Woodward and Bernstein and presented to them as the publisher of their "winning work" cited by the Pulitzer board in the award. Folks can feel free to decide for themselves how finely they want to split that particular hair.
https://www.pulitzer.org/winners/washington-postThe Pulitzer Prize for National Reporting was awarded in 2002 to "The Staff of the Washington Post" for a series of eleven articles cited as the "Winning Work" for the award. Bob Woodward was the sole author of four of the cited articles and coauthor on three others. As the lead reporter (as cited by the Washington Post) for the cited series and as part of the staff to which the award was given, I think most would consider that to mean he was a recipient of a Pulitzer Prize. Others may come to a differing opinion.
https://www.pulitzer.org/winners/staff washinton post 2002 national reportingFAQ #21 and #23
https://www.pulitzer.org/page/frequently-asked-questionsFrom the Washington Post Bio Page: "Bob Woodward is an associate editor of The Washington Post, where he has worked since 1971. He has shared in two Pulitzer Prizes, first in 1973 for the coverage of the Watergate scandal with Carl Bernstein, and second in 2003 as the lead reporter for coverage of the 9/11 terrorist attacks."
https://webcache.googleusercontent.com/https://www.washingtonpost.com/people/bob-woodward