In the Adams-Onis Treaty,Spain agreed to relinquish all of Florida to the United States on the condition that the United States RENOUNCE ALL CLAIMS on Texas and the lands beyond,all the way to the Pacific Ocean,FOREVER.
The cede land you mentioned did not happen until 26 years after Adams-Onis,under President Polk and the Mexican -American War,when Mexica was compelled to cede to the US all or parts of California,New Mexico,Arizona,Utah,Nevada,Colorado,and Wyoming.
Also,in ARTICLE V of Adams-Onis,Spain stipulated that"the inhabitants of the territories which HIS CATHOLIC MAJESTY cedes to the United States"were to be "admitted to the enjoyment of all privileges,rights,and immunities of the citizens of the United States"
The citizens of Florida at that time included Spanish,Seminoles,and Free African Americans,so in the summer of 1823.Florida Gov William Pope DuVall,with the help of James Gadsen,held a conclave at Moultrie Creek with more than 350 of Florida's non-white leaders.
It was there that,as Article I of Gadsen's text stated that they would be stripped of all human and property rights-"The undersigned chiefs and warriors,for themselves and their tribes...do cede and relinquish all claims or title which they may have to the whole territory of Florida"
It is interesting to note that both DuVall and Gadsen were involved in the formation of a "Golden Circle",and Gadsen is also known for the GADSEN PURCHASE of lower Arizona and New Mexico.