I say: This is the sound answer. The reply offered previously rests on the assumption that ʿĀʾishah never narrated the Prophet’s prayer on a reed mat, and, as you have seen, that assumption is unfounded.
On another occasion “he prayed on a reed mat that had turned black from long use.” This comes from the hadith of Anas b. Mālik. His grandmother Mulaikah invited the Messenger of Allah ﷺ for a meal; after he ate, the Prophet said, “Stand, so that I may lead you in prayer.” Anas continues: “I went to a reed mat of ours that had darkened from long wear, sprinkled it with water, and the Messenger of Allah ﷺ stood upon it. I and an orphan formed a row behind him, and the elderly woman stood behind us. The Messenger of Allah ﷺ led us in two rakʿahs and then departed.”
Mālik, al-Bukhārī, Muslim, and Aḥmad all relate it through Mālik, from Isḥāq ibn ʿAbdallāh ibn Abī Ṭalḥah, from Anas, and the wording just quoted is that of al-Bukhārī. Al-Nasaʾī and Aḥmad also transmit it through other chains from Isḥāq with a similar wording, adding: “and he prostrated on it.” An abridged version has likewise been reported, as mentioned under the first narration.
As for the report: “He prayed on the mattress upon which he and his family slept,” and the wording: “It was made of tanned leather and stuffed with palm fibers,” this is from the narration of ʿĀʾishah, may Allah be pleased with her. Aḥmad records it, saying: Ibn Numayr told us; Hishām related from his father, from her, that she said: “The Messenger of Allah ﷺ would pray while I lay crosswise between him and the qiblah, on the mattress.”