Is there a proper term for the origin on the timeline from which we count years? For example, we often have years in BC/AD or BCE/CE - what is this point of change from one to the other called. Origin, axis, omphalos, or what?
At the time when the so-called Christian era was introduced, the mathematical concept of zero was not known in Europe. Thus, the year before AD 1 is called 1 BC. However, in modern astronomical and mathematical usage “1 BC” is called the year “0”, “2 BC” is called “-1”, “3 BC” is called “-2”, etc. If you follow this convention you can carry out normal mathematical operations with year numbers. The starting point of the Christian calendar is 1 January AD 1; this is called the “epoch” of the Christian era Julian style. The equivalent date in the Gregorian calendar is 30 December year 0 (alias 1 BC). The epoch of the Gregorian calendar is thus the equivalent of 3 January year 1 (AD 1) Julian.
In chronology there is no year 0. The first year is year 1. The year before that is year -1. The origin of a time system is called its epoch.
What is the proper term for the “year 0”?
Year 1 BC.
There was literally nothing between year 1 BC and 1 AD. 0000 = 1 BC, and there was no separate "year 0 AD".
The reason why there was no "year 0 AD" between 1 BC and 1 AD is because in those days, the concept of zero was not introduced/invented yet.
Zero was invented in India in the sixth century, and was either transferred or reinvented by the Arabs by about the eighth century. The Arabic numeral for zero (0) did not enter Europe until the thirteenth century. Even then, it was known only to very few, and only entered widespread use in Europe by the seventeenth century.
If you want to convert the BC years into integers like the AD years, it makes 1 BC as
0, 2 BC as
-1, 3 BC as
-2, 4 BC as
-3, etc. This method is used by ISO 8601 and the Astronomical Year Numbering system.
There are some calendars which do have a year zero.
The calendar used by astronomers corresponds to the Gregorian and Julian calendars, but does not use AD or BC. Instead it has a year zero, coinciding with the Julian 1 BC, and before that years are negative. This continuity makes sense when you regularly work on time scales of billions of years.
Similarly, ISO 8601 designed for computer use has a year 0 corresponding to the Gregorian 1 BC.
The Shalivahana Śaka counts the first year of their calendar as year zero, similar to how we would describe someone in the second year of their life as being "1 year old".
Too long for a comment: The term epoch is also correctly used in computer programming
time() returns the time as the number of seconds since the Epoch, 1970-01-01 00:00:00 +0000 (UTC).
And gettimeofday() gives the same values with microsecond precision.
That is, when your higher level language or application gives you a date/time, it is calculating the numbers from the values described above.
after staring too much at raw timestamped datasets you even start to recognize numbers like 1590412116 (the number of seconds since the epoch right now)