There are many names being given to the Y2k phenomenon.
Millennium Bug, Century Bug, Year 2000 Computer Problem, Y2000 virus, Y2k and so forth.
The Third Millennium and the 21st Century begin on January 1, 2001.
So, strictly speaking, it is incorrect to associate the words "millennium" or "century" with a Year 2000 event.
Y2k is a 20th Century phenomenon.
It is not a virus.
"Bug", in the sense of a programming error, almost fits, but implies a very limited scope to the problem.
Because there are all sorts of things that contributed to the phenomenon; Failure to apply standards, Failure to provide for International usage, parochial date formatting preferences, Operating System design flaws etc, etc.
Use of the term "Millennium Bug" attempts to be cute and hip, but it implies total cluelessness and ignorance of the real nature of the problem on the behalf of the user. It is, to say the least, "uncool".
We are stuck with it, but understand that only Idiots will use the term.
"Year 2000", with the "Y" capitalised, is the most common usage.
"Y2k", the specific abbreviation coined to define "Year 2000 Problem" uses the greek suffix "k" (lower case) for "kilo" or "thousand".
Not to be confused with the "other" k suffix used in describing computer storage which is based on binary arithmetic (powers of 2) and means "1024".
Why do we need to be so pedantic about the name?
Well have you tried doing a Web Search for any of these terms? The problem is that "2000", "millennium" and "bug" are ambiguous. You pick up a tremendous amount of noise.
Only "Y2k" is specific. You know that you are not dealing with Millennium Madness or generic woffle about other things happening in or around the year 2000.
There are some subtle linguistic forces at work. It is absolutely correct to say "the first Y2k event occurred in 1972".
The hackles rise if one hears "the first Year 2000 event occurred in 1972" or "the first Millennium event occurred in 1972". It sounds wrong. In your mind you reject such statements even though they may be factually true. Subtle.
This entire topic is somewhat moot, as the damage is already done. Our sloppy human habits of speech and thought have once again proved to be a liability in dealing with the computer world.
So if you have the choice, use "Y2k". Otherwise "Year 2000". Let's leave "Millennium Bugs" to the politicians and other clueless dome-building party-animal Low Life.