You would likely start from the center of your galaxy, then pick a culturally relevant star near the outskirts, in our case Sol, and call that, well, something (Solbound?) Then work your right angles from there. You could pick other cultural stars in the other directions too.
Most galaxies are pretty flat, so you’d probably have to reference a conveniently positioned other galaxy for your up and down.
Whenever I think about this article, I think about how they could not have possibly known how emissions would grow, and they were perfectly reasonable to frame it this way. And if things stayed at that rate, we would have been able to do something about it so easily when we started getting worried