Is it mentioned in Sahih al-Bukhari that the Prophet (peace and blessings be upon him) only prayed on earth or similar, or that he would take a piece of cloth, place it on the earth, and then pray on it?

It is not mentioned in Sahih (definition) Al-Bukhari that the Prophet, peace and blessings be upon him, would only pray on earth. What is mentioned is the Prophet's request to Aisha to hand him "al-Khumra," which is a small prayer mat or something placed under the face during prostration. It is permissible to prostrate on a mat, on the folds of a turban, on the edge of a garment, or on the ground. However, it is disliked to dedicate only the forehead for prostration, as this was not reported from the Prophet, peace and blessings be upon him. Rather, what was reported was placing the "Khumra" under the entire face.