The redshift that is referred to is not a Doppler redshift, but a cosmological redshift. The difference is that the former is caused by the source moving through space, while the latter is caused by the "stretching" of the wavelength of the light as it travels through space.
The cosmological redshift happens gradually along its journey, and hence it is a measure of the distance to the source. But since traveling through space takes time, it is also a measure of the lookback time to the source — that is, the time that has passed since its emission. This notion of the term can thus be used as a timeline for phenomena in the Universe, and perhaps somewhat confusingly, it is sometimes used this way even when referring to local phenomena.
For instance, Earth was formed 4.54 billion years ("Gyr") ago. If some unrelated galaxy emitted light at the same time, and if that light reaches us today, then that galaxy must be at a particular distance$^dagger$. During its journey it has been redshifted to $z simeq 0.42$. Thus we may say that Earth was formed at redshift $0.42$.
Similarly, stars (in the Milky Way or elsewhere) that are said to have formed at redshift $z=10$ and $z=15$, can also be said to have formed 13.3 Gyr and 13.5 Gyr ago, respectively.
In the figure below, I plotted the relation between cosmological redshift and lookback time (assuming a Planck 2015 cosmology). Now hug each other, Rob and NotSoSN.
$^dagger$5.4 billion lightyears, actually. Not just 4.54 billion lightyears, because space has expanded in the meantime.
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