When did the popular idea begin that the 21st century would commence on 1st January 2000?

When did the popular idea begin that the 21st century would commence on 1st January 2000?

One can only speculate the when, but the likely reason is when writing dates with Roman numbers became uncommon.

People then often forgot that Roman numbers had no zero.

Since the Anno Domini calendar was introduced at a time when Roman numbers were common place (525), it started with year 1 and not 0 (for which there was no digit).

The last year of the first century (each century being 100 years) was therefore 100. The first year of the second century 101.

Roman numerals - Zero.
"Place-keeping" zeros are alien to the system of Roman numerals - however the actual number zero (what remains after 1 is subtracted from 1) was also missing from the classical Roman numeral system. The word nulla (the Latin word meaning "none") was used to represent 0, although the earliest attested instances are medieval. For instance Dionysius Exiguus used nulla alongside Roman numerals in a manuscript from A.D. 525.

The Anno Domini dating system was devised in 525 by Dionysius Exiguus to enumerate the years in his Easter table.

Year zero - Wikipedia

A year zero does not exist in the Anno Domini (AD) calendar year system commonly used to number years in the Gregorian calendar (nor in its predecessor, the Julian calendar); in this system, the year 1 BC is followed directly by year AD 1.
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1-to-0 decade.
A rarer approach groups years from the beginning of the AD calendar era to produce successive decades from a year ending in a 1 to a year ending in a 0, with the years 1–10 described as "the 1st decade", years 11–20 "the 2nd decade", and so on; later decades are more usually described as 'the Nth decade of the Mth century' (using the strict interpretation of 'century')

Start and end of centuries.
According to the strict construction, the 1st century AD began with AD 1 and ended with AD 100, the 2nd century spanning the years 101 to 200, with the same pattern continuing onward. In this model, the n-th century starts with the year that ends with "01", and ends with the year that ends with "00"; for example, the 20th century comprises the years 1901 to 2000 in strict usage.

This is because ancient Roman counting mentality. Latin language has "inclusive counting" features. Keep in mind that historical use of century counting was created by Renaissance latin-speaking European scholars.

Because the concept of the "21st century AD" is less useful than that of the "century of the 2000s" (or 2100x or whatever). It's entirely arbitrary.

To an excellent first approximation the only interesting thing about the starting date of the 21st century is bragging rights: "I know there was no Year Zero. Nyah!" It has no religious significance (even if it mattered exactly when He was born, Christ wasn't born in the Year 1.) It has no geological or meteorological significance. Nor astronomical, nor geopolitical, nor legal. (Possibly astrological?) It barely has any historical significance since AD dating wasn't used in the ancient world.

Because we're people and as such have a unremitting lust for round numbers, we mark anniversaries that are round when expressed in base 10 (and why has that any special significance?) so they end in zeros or are simple fractions of numbers which end in zeroes.

For most people the date odometer turning over from --99 to --00 feels a lot more significant than the count-of-years-elapsed-since-the-Year-One passing --99.

(It's true that people speak of things like the "nineteenth century" but I suspect most people, if they have to think about what years that was, follow the algorithm "Well, let's see, it ought to be the century of the 1900s, but I gotta remember to subtract one. Ok, the century of the 1800s." If it weren't for the widespread use of things like "the Sixteenth Century" to denote historical eras, the distinction would be limited to the occasional trivia contest.)