I have read of this process many times, but I don't think I know the term specifically for the moment when hydrogen fusion begins. What is this moment called?
Ignition or more fully Stellar ignition is the term you want.
It is known as the nuclear turn-on in the Hertzsprung-Russell diagram; the opposite of the nuclear turn-off when hydrogen is exhausted in the core.
In actual fact there is no "moment" when the star begins to fuse hydrogen, it is a gradual process.
For example, the models of Siess et al. (2001) suggest that a solar mass pre main sequence star gets 0.01% of its luminosity from hydrogen fusion at 2 million years old, 1% after 16 million years, but it is not until it is 56 million years old that 99% of the luminosity comes from hydrogen fusion.
The onset of hydrogen fusion for a star (i.e. when it enters the main sequence) is called the zero-age main sequence (ZAMS). Conversely, the terminal-age main sequence (TAMS) refers to the point a star stops fusing hydrogen (and by definition leaves the main sequence); the time between the ZAMS and TAMS is called the main sequence lifetime. On the Hurtzsprung-Russell diagram, several stars at their ZAMS are grouped into a line:
Some astronomers consider the ZAMS to be the "birth" of a star, in the sense that the object is no longer a protostar and meets the criteria of a star (source). These astronomers thus consider a star's age to be the time between now and its ZAMS. However, this is simply a matter of terminology, as others consider the formation of the star to be the moment when its molecular cloud collapses.