The question “What was a person thinking of as ‘today’?” in the Eastern Empire, as I asked over here on History SE, can have several possible answers depending on the era we live and the general circumstances at play in the imperial realm.
The simplest idea that the Western world has of the Ad Urbe Condita (from the original founding of Rome) is mostly an earlier, Principate, fiction that was used more commonly for ‘official’ dates than accurate timekeeping. Marcus Terentius Varro’s work–the author of the presently accepted calculation for the founding of the City–was accepted as gospel by Claudius for propaganda. Hence, it is unlikely many people in the empire ever thought of their present day in terms of how long after the founding of the City it was, lest it was a celebration of some kind, and I have no reliable information of these being continued in Byzantium/Constantinople.
The other early form of timekeeping was consular offices. In the Republic, it was common for years to be known as the “Year of Consul 1 and Consul 2”, in imitation of regnal years. Justinian I, however, abolished the practice of annual consuls. With the influx of repeating consular years (‘the first consulship of …’, etc), this method must have been more for recordkeeping rather than timekeeping in ordinary life. Similarly, when consuls were not appointed, the years were given in terms of how soon after an established consulship these took place. I cannot imagine many people thinking in these terms either, especially given the following few options.
The official calendar of the Empire (Etos Kosmou) between 988 (the 28th Year of Basileios II’s reign) and 1453 was the Creation Era, dated backwards to start at 1st September, 5509 BC. While 988 is when it was adopted by the Imperial government, earlier usage for religious purposes within the church was common ever since the 7th century. Local offshoots and earlier versions of the Etos Kosmou, such as the Alexandrian Era, existed at times, but would not have been as common throughout the empire (not to mention that Alexandria was lost forever in 641 AD, not including the occasional reconquests in the decades after).
Three of the more important methods of timekeeping have not yet been covered. These are the Julian calendar, regnal years (mentioned briefly above in relation to consular years), and the Indiction. The regnal years clearly must have been an important part of most peoples’ lives within the empire, especially as they followed previous Hellenic traditions of the eponymous archon (of Athens). Therefore, I would say foremost that most people always knew in what year of their Emperor’s reign they lived in.
This answer requires qualification: after 988, with an official reckoning adding power to the Church’s, it is not impossible that many people thought in both systems, but especially in the Church’s version. This is likely to be the case especially in periods where the Eastern court politics saw a variety of people take the throne in quick succession, in which case most provinces may have even been quite unaware of any changes in leadership. This is additional to Justinian I making the use of regnal years mandatory in 537 AD.
The Julian calendar would have been more common in some provinces, no doubt, especially due to the prevailing church influence. Hence, it is not unlikely that especially in the 4th to 7th century, plenty of people would have reckoned their time (year at least) based on Gaius Julius Caesar’s re-alignment of the traditional Roman calendar. However, it seems this gradually fell out of touch both with the increasing use of the Etos Kosmou as well as regnal years and the Indiction cycle gaining traction.
The Indiction was a 15-year cycle which also began on September 1st. Again it was Justinian I who decreed that all documents must be dated in this system. Indeed, based on the nature of this system, I feel that the vast majority of people would have been most familiar with this system. Even if the emperor’s name had changed, the tax collector would probably arrive on time. Hence, I think that for the duration of the Eastern Empire, this would have been the most likely answer to get from the majority of people, the Etos Kosmou being the second at least after the 10th century.
This is a copy of my own answer on History SE; an answer which I put up as I felt the present options there did not sufficiently answer my query. In a hopeful future, I will carry out more research on this topic to determine better sources and a more consistent narrative.