It was not made in 1804, but it was one of the first 1804 dollars made. In 1834, the U.S. State Department asked for a few sets of coins to be made for diplomatic purposes. This was one of those sets and the most famous. Since the mint had not made a silver dollar since 1804, but dated 1803, it made a dollar for this set, dated 1804 with the last design that had been used. If you own a Red Book this is one of the “original 1804 dollars.” Although the diplomat who was supposed to deliver it died, it eventually made its way the King of Siam. One of the subsequent owners was Anna Leonowens who with governess for the king’s children. She was immortalized in the Rogers and Hammerstein musical, The King and I. This 1804 dollar is one of the finest examples known. Fortunately it has been kept with the set. The set continues 1834 Proof coins. The $10 gold piece in it is also dated 1804 because the last of those coins were also made that year. Since there were 1804 $10 gold pieces made that year, it is not as famous.
The one in post #13 is one of the 2 surviving sets you mention. It, and the other surviving set that was slabbed can be seen in this thread - https://www.cointalk.com/threads/1868-experimental-aluminum-proof-set.311006/ I had the luck to be able to examine the slabbed set and photograph it at the ANA show back in '07. I've posted about it a few times in various threads here on the forum.
I've told Ron Landis that he should have made an 1834 capped bust dollar and an 1834 classic head bust $10 to properly "finish" those proof sets, based on the 50c and $5 designs, respectively, but scaled up in size and amount of detail.
That would be the right think to do. But make at least a 1000 of them. Only one person can own a 1of1.