The challenge when trying to answer questions about history is that all we can rely on is evidence that has come to us (such as texts or archaeological evidence) and often it is not 100% clear what happened.
In any case, there are some texts that say that 'Umar ibn al-Khattab ordered that the line "prayer is better than sleep" be added to the Fajr adhan. For instance, in al-Muwatta' (a prominent early Sunni source), it is said that the mu'adhdhin came to Umar ibn al-Khattab at fajr time and found him sleeping, so he told him, "Prayer is better than sleep" (to tell him to get up), and then 'Umar ibn al-Khattab ordered that it be added to the adhan. (See the relevant chapter in the book on salat in the Muwatta')
However, some people, especially Sunnis, hold a different view.
I think it is reasonable to say that all Muslims who do the adhan try to do the adhan in the way they think is correct and the way the Prophet (S) intended it. Shi'i fiqh does not prescribe "prayer is better than sleep" in the adhan because the dominant Shi'i view is that this is was not how the Prophet (S) instructed Muslims to do it. At the end of the day, Allah judges on intentions, and we do our best to follow the Sunnah!