The link you provided was to Stephens' address to the Georgia Secession Convention and not the Cornerstone speech. Are you saying that speech was incorrectly reported too?
No, if I posted the wrong link, that’s my error.
“The speech was first reported in the Savannah Republican newspaper, with this version appearing in a collection of Stephens’s papers, edited by Henry Cleveland and published in 1866.”
https://encyclopediavirginia.org/entries/cornerstone-speech-by-alexander-h-stephens-march-21-1861/
Note “a collection of Stephen’s papers.”