The federal reserve is indeed part of the government. It has separations from the executive branch but it is indeed part of the government.
The same website you linked to has an FAQ page dedicated to clearing up this misconception: https://www.federalreserve.gov/faqs/about_14986.htm
> The Federal Reserve System is not "owned" by anyone. Although parts of the Federal Reserve System share some characteristics with private-sector entities, the Federal Reserve was established to serve the public interest.
> Some observers mistakenly consider the Federal Reserve to be a private entity because the Reserve Banks are organized similarly to private corporations. For instance, each of the 12 Reserve Banks operates within its own particular geographic area, or District, of the United States, and each is separately incorporated and has its own board of directors. Commercial banks that are members of the Federal Reserve System hold stock in their District's Reserve Bank. However, owning Reserve Bank stock is quite different from owning stock in a private company. The Reserve Banks are not operated for profit, and ownership of a certain amount of stock is, by law, a condition of membership in the System. In fact, the Reserve Banks are required by law to transfer net earnings to the U.S. Treasury, after providing for all necessary expenses of the Reserve Banks, legally required dividend payments, and maintaining a limited balance in a surplus fund.