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".