That is a good example of sudden conversion of paul and umar. But there is a lot of difference. Umar converted during the lifetime of the prophet. He was going to kill the prophet but was informed of the conversion of his sister.
So he went to her home and beat up the sister and brother in law. Then sis ter spoke out in favor of Islam and told Umar that he could do what he liked. They will not leave Islam.
That incident of Umar is totally different to that of obscure incident of a flash on the road to damascus for paul. For Umar prophet was alive and accepted him. For paul there was nobody. Umar suffered all the hardships of persecution and difficulties for many years with the prophet. Did paul even see Jesus for one day.
What do you mean by “for Paul there was nobody”? Christ’s Church was present to accept or refuse new followers wishing to join the Christian community. If you read the Acts of the Apostles, you will find out that Paul saw Jesus on many occasions. The only thing you fail to understand is that who appeared to Paul on the way to Damascus was the risen Jesus, no one else.
For the pure reason of playing the skeptical, I shall ask you a few questions. You say Omar was authorised by Mohammad himself. What if that story of Omar’s conversion during Mohammad’s lifetime is but a lie? What if that conversion story - the one you consider totally different from the one in Christianity - is a fake one concocted during Omar’s period? What if Omar changed his mind after making his way to the throne and determined to change Mohammad’s message after his death?
Besides, I can find no verse in the Koran that relates and endorses Omar’s sudden conversion. Where in the Koran does Allah or Mohammad authorise Omar as a sincere believer and Mohammad’s successor?

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Then Umar followed the teachings of the prophet and the Quran. Paul did not have Jesus and did not have any written book with him. Besides his teachings are 180 degree opposite to what Jesus had been teaching. Good-Bye to paul.
Paul did see Jesus indeed! This is one of the main differences between Christianity and Islam. In Islam even conversion stories are expected to have occurred during Mohammad’s lifetime since Mohammad’s death snatches him away from this world whereas in Christianity Jesus’ death is followed by His resurrection, through which He keeps tending His flock and protecting His message until the day of judgment:
Matthew 28:20 And remember,
I am with you, always, to the end of the age.”
Peace,
Angelos N. B.