Population density of stars within $13$ light-years of us
The criteria for sample selection are admittedly not made overly clear in the preprint itself (Wysoczańska et al.). It appears that the authors started with a list based on stars that were previously believed to have come close to the Sun at some point in the past or future. That list was then augmented based on data from Gaia's second data release, to get a total of 820 objects, and the motions of those stars were then integrated backwards and forwards in time from the present, based on a model of the galaxy's gravitational field.

Here's where that "over 600 stars" came from. The authors ended their integration for a given star when either (i) the star went more than 3 kpc from the Sun or (ii) the time reached 50 million years in the past or the future. With those cutoffs in place, the astronomers found that during that timespan (50 million years before or after the present), 647 stars were within 4 parsecs of the Sun at some point in time. That's the figure Fox is referring to.