Al-Masjid al-Aqsa is the name for the whole of the place of worship built by Sulaymaan (peace be upon him). Some people started to give the name of al-Aqsa to the prayer-place which was built by Umar ibn al-Khattab in front of it.
Praying in this prayer-place which Umar built for the Muslims is better than praying in the rest of the mosque because when Umar conquered Jerusalem there was a huge garbage dump on the rock, since the Christians wanted to show their scorn for the place towards which the Jews used to pray. Umar issued orders that the filth be removed. He said to Ka’b [al-Ahbar]: “Where do you think we should build a place of prayer for the Muslims?” Ka’b replied, “Behind the rock.” Umar said, “O you son of a Jewish woman! Are influenced by your Jewish ideas! Rather I will build it in front of it.”
Hence when the imams of this ummah entered the mosque, they would go and pray in the prayer-place that was built by Umar.
With regard to the Rock, neither Umar nor any of the Sahabah prayed there. There was no dome over it during the time of the Rightly-Guided Caliphs. It was open to the sky during the caliphate of Umar, Uthman, Ali, Mu’awiyah, Yazeed and Marwaan… The scholars among the Sahabah and those who followed them in truth did not venerate the rock because it was an abrogated qiblah… rather it was venerated by the Jews and some of the Christians.”
(Majmoo’at al-Rasaa’il al-Kubra, 2/61)
[Umar denounced Ka’b al-Ahbar and called him the son of a Jewish woman because Ka’b had been a Jewish scholar and rabbi. When he suggested to Umar that he should build the mosque behind the rock, it was out of respect for the rock so that the Muslims would face it when praying, and veneration of the rock was part of the religion of the Jews, not the religion of the Muslims. – IslamQA]