Is a person's worship from Fajr prayer until sunrise considered an 'Umrah, and does that expiate the sins between the previous 'Umrah and this alleged "Umrah"? And does the reward for that differ between Ramadan and other months?

There is no evidence that reciting the Quran from Fajr until sunrise is equivalent in reward to an umrah. Rather, what is narrated in this regard is a weak hadith, which some have deemed hasan (good), and it is: "Whoever prays al-ghadah (Fajr) in congregation, then sits remembering Allah until the sun rises, then prays two rakahs, will have a reward like that of a complete, complete, complete Hajj and umrah." This hadith indicates the virtue of sitting and remembering Allah after Fajr prayer in congregation until the sun rises, then praying two rakahs thereafter. Allah's bounty is vast, and whoever strives to attain the reward mentioned in this hadith is hoped to receive Allah's forgiveness. The virtue of this sitting in Ramadan is greater than at other times, as the rewards for acts of obedience are magnified and their recompense multiplied in Ramadan.