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I am going to post an answer of my own (which I also copied to my blog, on the attached link which references back to here -- both the question and the answer), using both references from my post here as well as the other present answer. I am not accepting the other post as the answer I accepted because I don't think they have answered my actual question as opposed to posting a series of links to relevant but also irrelevant topics. I highlighted as much in a comment last week, but no substantive edits have taken place, and I am therefore thinking this will provide a more accurate answer.
The question "What was a person thinking of as 'today'?" in the Eastern Empire 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.
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Many different local calendars were used in the eastern and western parts of the Roman empire. The Julian calendar introduced about 44 BC was the official calendar of the Roman republic and empire and was used by Roman citizens, the army, and the imperial administration.
https://en.wikipedia.org/wiki/Julian_calendar
The Julian calendar could be used with any "era" or year count, but if it didn't use a starting year that made the leap years the same as in the official version then the dates could be off.
The official Roman year began on January First, but the Julian calendar and other calendars could start the year at any date. For example, medieval England used the Julian calendar, but started the year on The Feast of the Annunciation, March 25, from 1155 to 1752. March 25 was the first day of the year, and March 24 was the last day of the year.
It was common to call the year by the names of the regular consuls of the year. For example, 59 BC was called the consulship pf Caesar and Bibulus, but since Gaius Julius Caesar was more dominant it was often jokingly called the consulship of Julius and Caesar. So historians had to keep long lists of the successive consuls.
https://en.wikipedia.org/wiki/Roman_consul#Consular_dating
During the empire it was common to use the regnal year of the emperor or emperors, and Justinian made it mandatory in 537 AD.
https://en.wikipedia.org/wiki/Julian_calendar#Year_numbering
Romans sometimes counted the years ab urbe condita or anno urbis conditae, AUC, since the founding of Rome. Emperor Claudius made the calculation of Marcus Terentius Varro official, officially dating the foundation to 21 April 753 BC.
https://en.wikipedia.org/wiki/Ab_urbe_condita
Hellenistic Greeks often dated events by Olympiads, the first Olympiad being from 776 BC to 772 BC.
Diodorus Siculus dates the Persian invasion of Greece to 480 BC: "Calliades was archon in Athens, and the Romans made Spurius Cassius and Proculus Verginius Tricostus consuls, and the Eleians celebrated the Seventy-fifth Olympiad, that in which Astylus of Syracuse won the stadion. It was in this year that king Xerxes made his campaign against Greece."
Jerome, in his Latin translation of the Chronicle of Eusebius, dates the birth of Jesus Christ to year 3 of Olympiad 194, the 42nd year of the reign of the emperor Augustus, which equates to the year 2 BC.
https://en.wikipedia.org/wiki/Olympiad
Hellenistic Greeks often used the Seleucid era, or year of the Greeks, dating from 312/11 BC.
https://en.wikipedia.org/wiki/Seleucid_era
Christian chronologists often dated events from the Crucifixion of Christ or from his incarnation. The modern AD/BC system came into use in western Europe during and after the reign of Charlemagne.
The 15 year indiction cycle began for tax purposes about 300 AD. In a few decades non tax documents began being dated by the indicition cycle. In 537 AD emperor Justinian decreed that all dates must include the indiction.
https://en.wikipedia.org/wiki/Indiction
Christians also often dated events from the biblical creation (anno mundi or Etos Kosmou) of the world. Scholars proposed - and often used in their chronicles - tens and even hundreds of different dates for the creation of the world. I myself once calculated several different dates of Creation.
Most dates for the Creation are either within a few centuries of the date of Creation in the Hebrew Calendar, 6 October 3761 BC, or else within a few centuries of the date of Creation in the eventual Byzantine Era adopted in 988 AD, dating Creation to 1 September 5509 BC. The Byzantine Era was first adopted by the Sixth Ecumenical Council in 680/81 AD and first officially used in the Quinisext Council of 692 AD or 6200 AM.
Thus the Hebrew year 5777 began on 2 October 2016, while the Byzantine year 7526 should begin on 1 September 2017.
Professor Fr. Arsenius John Baptist Vuibert (S.S.), a 19th-century historian, observed that Biblical Chronologies are uncertain due to discrepancies in the figures in Genesis and other methodological factors, accounting for hundreds of different chronologies being assigned by historians. In the case of the Fathers of the Sixth Ecumenical Council, who assigned 5509 BC. as the date of the creation of man, he writes that it was in response to the emperor's wishes to fix an era or convenient starting point for historical computation. Therefore, it was a decision of mere historical convenience, not respecting either faith or morals, which are what is truly of intrinsic value in the Scriptures.[43] Having made this disclaimer, he settles on the Benedictine Chronology of 4963 BC for the purposes of his history.
According to the ninth edition of the EncyclopΓ¦dia Britannica, regarding the so-called Era of the Creation of the World, Alphonse Des Vignoles asserted in the preface to his Chronologie de lβHistoire Sainte (Chronology of Sacred History, Berlin 1738), that he collected upwards of two-hundred different calculations, the shortest of which reckons only 3483 years between the creation of the world and the commencement of the vulgar era and the longest 6984. The so-called era of the creation of the world is therefore a purely conventional and arbitrary epoch, for which the very nature of the case discussion is hopeless labour.
https://en.wikipedia.org/wiki/Anno_Mundi
https://en.wikipedia.org/wiki/Byzantine_calendar
If there is ever a revived Roman Empire, it should use a triple year count to honor the original Roman Empire, the Byzantine Empire, and the Holy Roman Empire, so that the year might be given as Etou Kosmou 7609, AUC 2853, and AD 2100.