The main sequence stage for a star begins after it stars burning hydrogen into helium. This is often referred to as the zero age main sequence.
This is confusing since stars have already lived several million years in their pre-main sequence stage.
Where does this come from? Is it a historical designation that just stuck?
The zero age main sequence (ZAMS) is the theoretical locus of points in the Hertzsprung-Russell diagram where the luminosities of young stars (at a range of masses) are mostly supplied by the nuclear reactions that have begun in their cores.
Prior to this, the objects are known as pre-main sequence stars and most of their luminosity arises from the release of gravitational potential energy as they contract.
The exact definition (whether it is 99% or 90% of the luminosity) does vary, but since the nuclear reactions ramp up quite quickly this doesn't materially affect any quantitative analysis.
In any case, the age of a star is not measured from the ZAMS. The reason for this is that what is of physical interest is the time since the star formed and that must include the pre-main sequence lifetime. Whilst this is very short (maybe 1 million years) for stars more massive than the Sun, and certainly shorter than the total main sequence lifetime, it can be much longer (100 million years) for a low-mass M dwarf. This is also the reason that the ZAMS is a theoretical construct. Stars born at the same time will never be all on the ZAMS at the same time. By the time low-mass stars reach the ZAMS, more massive stars will have evolved away from the main sequence!
As to the origin of the term, I suspect it comes from the early computer/analytic models of stellar evolution. There you would likely miss out the pre main-sequence phase and start your model with "zero age" at that point where it achieves equilibrium with the luminosity supplied by fusion and with a pristine He/H ratio in its core. Thus in reality, stars of different mass would have different ages at this point, but from the point of view of a theoretical modeller, it would be their starting condition and hence a "zero age".
The term "zero age Main sequence" is there only because during main-sequence evolution, hydrogen is turned to helium, changing the stellar structure a little (electrons are eaten up, light escapes more easily, so the luminosity rises for example). Yet all stars along that process are called "main-sequence stars." So the "zero-age" subset is simply a term for eliminating confusion about what the rising helium content is doing, it's not intended to be a statement about the actual age of the star. When astronomers talk about the age of a star, it's the age before the ZAMS, plus the age after the ZAMS. Usually the distinction doesn't matter, because main-sequence lifetimes tend to be much longer than pre-main-sequence lifetimes. This is convenient, because it is not possible to identify an exact moment when a star "first forms."
The situation is made even more unclear when you look at the formal definition of a "star," which is almost never used in practice but requires that the object be undergoing fusion. So if you believe that definition, the ZAMS is also the birth of the star, and its age is the age on the main-sequence. No one actually uses that poor language, because we do actually want to talk about the pre-main-sequence lifetime of a star! I suppose it would be a bit like counting the 9 months we are in the womb as part of our "lifetime," but this is the standard for what is done in professional astronomy (though not for human lifetimes).
This answer may be a bit speculative but still I thought to give it try.
A.S.Eddington The Internal Constitution of the Stars (Cambridge University Press, 1926) has probably been one of the most influential publications on stellar models and almost every physicist of that time working in that field will have read it.
On page 302 ff. of the book Eddington discusses the energy generation of the stars and its dependency on physical properties, like temperature and pressure, but on page 304 also on the age of the material. There he writes, referring to the consumption of material, and thus aging of the star:
The question what zero the age is to be reckoned from remains unanswered.
I’m not a native speaker and this sounds quite old style English to me, so I may be misinterpreting something here. But since every person working in the field of stellar models will have read this book, it seems quite plausible to me that Eddington here may have coined the term zero age.
Here’s the link to the page: https://archive.org/details/TheInternalConstitutionOfTheStars/page/n311
Honestly, I hadn't heard the term 'zero age main sequence' until this question came along. The Wikipedia page you linked mentions the zero age term and provides a link to an explanation.
Zero Age Main Sequence refers more to a specific point in a star's life. When determining the age of a star, astronomers will sometimes start counting from when it joins the main sequence (so it would be 'zero years old' when it does so). Before this time in this case, the object would be considered a protostar.
