and in 1 AD...did the people at the time consider it 1 AD?
no
The church set up the year 0, it was a monk called Dionysius Exiguus (lived around 500 ++ AD)who was asked by Pope John I to set out the dates for Easter.
Dionysius Exiguus declared Jesus was born,, on December 25 in the Roman year 753 and the new year started with January 1, 754(Roman Year) He later on called 754 for year 1 anno domini.(He chose the year in which Rome had been founded and determined, from the evidence known to him, that Jesus had been born 753 years later.)
These were his arguments (ohh I love this 😆 )
-Dionysius Exiguus and his contemporaries claimed that God created the earth on march 25th and since they believed Jesus to be perfect he had to have been conceived on march 25th(what logic minds they had)which led to his birth 9 months later on december 25th
December 25 was a favourable choice. In around 270 bc the Roman Emperor Aurelian declared December 25 a civic holiday in celebration of the birth of Mithras, the sun god. By 336 Christians countered by celebrating the birth of Jesus, the son of God, on December 25.
The Romans accepted a mix of religions and that mixing led to a mix of images of The Sun god and Son of God . And this mixing is believed to be the origin of Jesus with a halo
A pope called Gregory "cleaned" up the calendar in 1582 which led to that the calender lost 11 days.
Gregory also synchronized the calender with the movement of the sun and i think it was that October 4 became October 15 and to avvoid further adjustment leap years was introduced.
However Dionysius Exiguus dating of JC birthday conflict with the gospels of Matthew which claims that Jesus was born while Herod was king, which means Jesus would have been born in year 4bc or before