Generally by the reigns of kings, "In the sixth year of the reign of King Richard," etc. Rome started out as a Republic, so it used the names of the two ruling counsels for that year (they served one-year terms in office starting on New Year's day). So a Roman might say that an event happenned "during the counselship of Pompey and Crassus" etc. For a few things they used the date of the founding of the city; the Emperor Philip threw a huge celebration for what was calculated to be the city's 1,000 birthday (248 AD to us).

You're right that different people used different systems....I'm trying to remember exactly where I read this, I think it was somewhere in Mary Beard, but she cited an example from an ancient writer (Herodotus?) who described an event as occuring "In the 7th year of Emperor X's reign, in the spring that the Dictator Y took power Athens, and during the third year of the war between the Parthians and the Armenians..." etc. He was basically name checking the dating systems of several different cultures so that someone reading his work would be able to triangulate when something happened using the ones they were familiar with.

Dionysius Exiguus started using a system of reckoning based on his dating of the years since the birth of Christ in 525 AD. It became more widespread in terms of usage by the late 9th century but for a comparison point Portugal is regarded as the last western European country to adopt it in 1422. Russia lagged much later to 1700!

BC wasn't used used until after 700 in literature (Venerable Bede of Northumbria published his "Ecclesiastical History of the English People" )

So what about the Sumerian people. Didn't they "invent or were told" about the base 60 counting ? It was about 6,000 years or so go?