There is not a manual or handbook that tells you how someone counted a reign, or when the power of the next in line to throne officially began. All we can look at is convention and how it worked with each ruler, keeping in mind that different rulers were treated differently based on the manner in which they came to power and how their reigns were viewed by the public.
the existence of various calendars, lack of knowledge about customs
concerning the reckoning of the accession (part-) year, and especially
a period of partial coregency with Augustus exclude certainty1
In the case of Tiberius, Luke was writing for an audience - the early church in the Roman Empire - and would use the dating conventions of that audience so here we should defer to existing practices for Tiberius. This inevitably becomes a question of years or dates as no universal custom is known to have existed, nor is there a reason to believe the same standards were applied to all emperors. In terms of dates for Tiberius, there is debate about whether the coregency was reckoned from 11 AD, 12 AD, or 13 AD, with the majority traditional view being that it goes October of 12 AD:
On Oct 23, A.U.C. 765 = AD 12, he celebrated a triumph for his
military victories in Germany and Pannonia. Referring to this event,
Suetonius says that "the consuls caused a law to be passed soon after
this that he should govern the provinces jointly with Augustus and
hold the census with him." The date when Tiberius thus began to govern
the provinces jointly with Augustus was probably AD 12 [Suetonius, ed
J.C. Rolfe (LCL), vol 1, 323] although arguments have been presented
for putting it in AD 11 or 13[Holzmeister, Chronologia vitae Christi,
p 66].In this connection Tacitus describes Tiberius Nero as collega
imperii, "colleague in the empire" (Annals 1.3), and some consider him
joint emperor with Augustus from this time on [The Encyclopaedia
Britannica 22:176]
As to why reigns were expanded beyond how we would reckon reigns today from a purely historical view, imagine being taken in front of Tiberius in AD 20 - you would certainly declare his reign to begin from the earliest possible date. So it is about power.
For instance, we can look at numismatic evidence, and we don't see any coins commemorating Tiberius until after August AD 14 - but after that point, once he acquired power, it was conventional to date a longer reign than from AD 14 because this is what Tiberius himself promulgated. Augustus dies in August and Tiberius immediately appoints Gratus as Prefect of Judea to replace Rufus (who had been Augustus' appointee), and in the same year Gratus is already minting coins with Tiberius' image that have the LB inscription, meaning the second year of Tiberius' reign.
Because this is Tiberius' policy and not necessarily the view of everyone in the Empire, there are counterexamples as to regnal Imperial reckoning, but Tiberius appointed governors in Judea to support his claims. And this policy continued, so that in AD 15, Gratus mints an LΓ coin, and in AD 16 he mints an LΔ coin and in AD 17 he mints an LΕ coin and in AD 23 an LIA coin (11th regnal year) So there is absolute no doubt that in terms of Tiberius' own marketing, his reign begins in the fall of AD 12. Now perhaps there was a governor who was a holdover from Augustus in a different province that did not follow the official line and minted some other coin, but we do know what the policy was in both Palestine and Antioch, where Luke wrote.
And we have even more numismatic evidence to confirm this, as Gratus's replacement, Pontius Pilate, continued the policy printing "LΙϚ" (LIS) coins denoting the 16th year of Tiberius reign, LIZ the 17th, and LIH the 18th year in AD 28, and AD 29 respectively.
Prior to A.D. 29 most coins except those that Gratus initially minted
in A.D. 14 included an honorific for Julia Augusta up to and including
the LIS coin dated Tiberius' 16th year which was struck with "Empress
Julia" and minted in A.D. 28. But subsequent to Julia Augusta's death
in early A.D. 29, the LIZ and LIH coins minted later in A.D. 29 and
30, respectively, did not have her honorific, thus establishing
another synchronism resulting from reckoning Tiberius' years as
factual from his joint-governance with Augustus in A.D. 12.
I provided only two data points, from Gratus and Pilate official policies, but there is a lot more evidence to review - however the main point is that Tiberius wanted his reign to be reckoned from AD 12 - even though not everyone reckoned according to Tiberius' wishes, especially not every later historian. When interpreting regnal years, we must do so as a subject of the empire rather than as a historian looking from outside the empire.
Reigns, in the mind of a subject of the empire, depended on how the Emperor, once in power, presented his reign to the people, in terms of minting coins, making announcements, and publishing decrees. If he wanted to say his regency began in X years, then that's what would be promulgated and used in collations of eye-witness reports such as Luke's. The man on the street would listen to the proclamations that some work was completed in the Xth year of a reign, or look at the coins, and say "This is the third year of Tiberius" in AD 15, and then when reporting an eye witness event, he would say that in the third year of Tiberius something happened, and then these testimonies would be accumulated by Luke in his reporting, assuming that Luke based his reporting on contemporary accounts during the time period in question.
But at the same time, a more scholarly work written in Rome by a later historian might ignore Tiberius' reckoning. And something written by an enemy of Tiberius would shorten his reign. This explains why there is conflicting evidence.
So no handbook or algorithm exists that can answer this question for all emperors or written documents. It is just false to say "Romans used X". At best, you can say "One historian used X" and "People in another province used Y", etc. You need to treat things on a case by case basis to obtain data from how regency periods were presented to, and therefore viewed by, the public. Nor can you separate this from the power politics at play in order to give it a veneer of objectivity. This will disagree with how modern day historians calculate regnal reigns and is a good method of distinguishing contemporaneous testimony from later accounts.
Nolland, John. Luke 1:1–9:20. Vol. 35A. Word Biblical Commentary. Dallas: Word, Incorporated, 1989.
http://theos-sphragis.info/years_of_tiberius_timeline.html#coregent
