Who added the phrase "Salah is better than sleep" to the Fajar azan?

I am a bit confused. 1) someone mentioned that these phrases *salah is better then sleep* were added by umar ibn khattab after Prophet PBUH time and somewhere else it is mentioned with references that hazrat Bilal R.A was adviced by Prophet PBUH to add these phrases only in fajar azaan and not in all the azans like he used too. So I want to know 1) if Prophet PBUH was alive and well when these phrases were added in Azan 2) and that were these phrases added by Prophet SAW himself due to Hazrat Umar bin Khattab incident.?? >> (referening to the incident I read where someone said these phrases were added by Umar bin Khattab after Prophet PBUH time when: once when he R.A himself slept through fajar azan and azan reciter came to wake him up and said these phrases and was then asked to add these in every fajar azan for the muslims.) 3) or were these phrases asked to be added by Umar Bin Khattab (in relation to above mentioned incident) after Prophet SAW's time.??