Explanation of the disagreement over whether the one to be sacrificed was Ismail or Ishaq (peace be upon them)

It is narrated from Sayyiduna Abdullah bin Abbas (may Allah be pleased with them both) that he said: "The Dhabih (the one who was to be sacrificed) is Isma'il (peace be upon him)."
This hadith is sahih (authentic) on the condition of the two Shaykhs, but they did not record it.

It is narrated from Sayyiduna Abdullah bin Umar (may Allah be pleased with them both) concerning the saying of Allah Almighty, "And We ransomed him with a great sacrifice" [Surah al-Saffat: 107], that he said: "He is Isma'il (peace be upon him), on whose behalf Ibrahim (peace be upon him) slaughtered the ram."

It is narrated from al-Sunabihi (may Allah have mercy on him) that he said: We were present in the gathering of Sayyiduna Mu'awiyah bin Abi Sufyan (may Allah be pleased with them both) when the people began discussing Isma'il and Ishaq, the sons of Ibrahim (peace be upon them both). Some of them said: "The Dhabih is Isma'il," and some said: "Rather, the Dhabih is Ishaq." Mu'awiyah (may Allah be pleased with him) said: "You have come to one who is well informed. We were with the Messenger of Allah (peace and blessings be upon him) when a Bedouin came and said: 'O Messenger of Allah, I have left my land in a state where the earth is dry and the water has dried up; livestock have perished and families are being ruined. So favour me with some of what Allah has granted you as spoils, O son of the two Dhabihs (the two who were to be sacrificed).' The narrator says: The Messenger of Allah (peace and blessings be upon him) smiled and did not object to him. So we said: 'O Commander of the Believers, who are the two Dhabihs?' He said: 'When Abd al-Muttalib was commanded to dig Zamzam, he vowed to Allah that if this task of his was made easy he would slaughter (sacrifice) one of his sons. So he brought out his sons and cast lots between them, and the lot fell upon Abdullah. Abd al-Muttalib intended to slaughter him, but his maternal uncles from Banu Makhzum stopped him and said: Please your Lord and ransom your son.' Mu'awiyah (may Allah be pleased with him) says: 'So he ransomed him with a hundred she-camels. So he is the (first) Dhabih, and the second is Isma'il (peace be upon him).'"

It is narrated from Sayyiduna Abdullah bin Abbas (may Allah be pleased with them both) that he said: "The one who was ransomed (that is, the Dhabih) is Isma'il (peace be upon him). The Jews have claimed that he is Ishaq (peace be upon him), but the Jews have lied."

It is narrated from Sayyiduna Abdullah bin Abbas (may Allah be pleased with them both) that he said, concerning the one whom Allah Almighty ransomed with a great sacrifice: "He is Isma'il (peace be upon him)."
This hadith is sahih (authentic) on the condition of the two Shaykhs, but they did not record it.

It is narrated from Muhammad bin Ka'b al-Qurazi (may Allah have mercy on him) that he said: "The one of his two sons whom Allah Almighty commanded Ibrahim (peace be upon him) to slaughter was Isma'il (peace be upon him), and we find this in the Book of Allah, in the account of Ibrahim (peace be upon him) and the command given to him to slaughter his son, that it was Isma'il (peace be upon him). The proof of this is that when Allah Almighty completed the account of the one of Ibrahim's two sons who was to be slaughtered, He said: 'And We gave him glad tidings of Ishaq, a Prophet from among the righteous' [Surah al-Saffat: 112]. Then in another place He says: 'So We gave her glad tidings of Ishaq, and after Ishaq, of Ya'qub' [Surah Hud: 71], that is, glad tidings of a son and then of a grandson. So Allah Almighty could not command the slaughter of Ishaq when He had a promise concerning him which He had made to him (that Ya'qub would be born from him). So the one whom he was commanded to slaughter could be none other than Isma'il."