Commems,
I agree with you that the first two official US Commemoratives were those issued by the US Mint to mark the Centennial in 1876. However, they were not the only official Centennial Commemoratives. There were four distinct issues, nineteen different medals in the various alloys, and at least eight more specimens if you collect the die varieties.
From my soon-to-be-published book on the Centennial medals of 1876:
On February 12, 1873, Congress authorized the striking of National Commemorative Medals by the United States Mint. The first four issues authorized under that Act celebrated the impending Centennial of the United States of America.
The first National Commemorative issue authorized, struck, and sold were the medals in two designs and sizes for the U.S. Centennial Exposition to be held in Philadelphia. They were authorized by Congress on June 16, 1874, struck later that same year, and first offered for sale on March 24, 1875.
The second commemorative issue celebrated the Centennial of the Battle of Lexington, which took place on April 19, 1775. The first order of 2 gold, 25 silver, and 200 bronze medals was rushed into production on April 15, 1875, in order to be available for the Centennial celebration in Lexington, Massachusetts, on April 19th and 20th, 1875.
The third issue commemorated the Centennial of the Mecklenburg Declaration of May 20, 1775. The first reverse die produced by the Mint drastically misspelled the name of the county as "Mechlenburgh," which delayed production until May 5, 1875. The medals were introduced to the public on the 19th and 20th of May, 1875, at the Mecklenburg Declaration Centennial Celebration in Charlotte, North Carolina.
The fourth and last set of Centennial medals issued under the Act of 1873 was the Nevada Exposition medal, commissioned to be sold as a Centennial Exhibition souvenir at the California and Nevada State Building. Production started on June 20, 19876, and ended on November 11, 1876. According to a certificate signed by A. Loudon Snowden, U.S. Mint Coiner, that accompanied the first 147 medals, "the Nevada Exposition medals... are made of pure silver, crushed from Nevada ores, at the Nevada quartz mill in the Centennial Exposition grounds, and subsequently refined at the United States Mint.