This article indicates that Umar I would refer to Umar ibn al-Khattab. It gives a tale of Umar ibn al-Khattab offering marriage to his son to a particularly honest woman who would eventually become Umar II grandmother. It then gives a brief biography of Umar II explaining why he is so revered to this day by Muslims regardless of sect.
While I was looking through a couple of other articles on the subject I also found out a few more interesting facts:
-Umar ibn al-Khattab once said that a descendant of his would be scarred and would right injustices during his rule. Umar ibn Abdul-Aziz was scarred on his face during his childhood by an animal and was widely seen as a just ruler.
-There was an Umayyad practice of cursing Ali’s name at the end of sermons. In an attempt to reconcile the Muslim factions, Umar ended this and replaced it with a verse from the Quran: “God commands you to practice justice, enjoins you to help and assist your kin and He forbids obscenity, evil or oppression, so that you may remember Him.”
Also, he is referred to as the either the fifth or sixth rightly guided caliph in several of the sources.