According to this website, the accuracy of the website you linked to is not entirely accurate.
They state here that
The calculations in the NOAA Sunrise/Sunset and Solar Position
Calculators are based on equations from Astronomical Algorithms, by
Jean Meeus. The sunrise and sunset results have been verified to be
accurate to within a minute for locations between +/- 72° latitude,
and within 10 minutes outside of those latitudes.
It later states that
The effects of the atmosphere vary with atmospheric pressure, humidity
and other variables. Therefore the solar position calculations
presented here are approximate. Errors in sunrise and sunset times can
be expected to increase the further away you are from the equator,
because the sun rises and sets at a very shallow angle. Small
variations in the atmosphere can have a larger effect.
The +/-72° line is roughly near the arctic circle on both sides of the globe, so for most Halachic zmanim calendars, the margin of error would not be more than a minute.
They also provide an assumped atmospheric refraction level, and some other mathy stuffs.
Now, as to the halachic websites which calculate the zmanyim to the second, they write at the bottom of the list:
**The times for sunrise and sunset have been computed assuming that the horizon is clear of obstructions (see Accuracy).
Later in the Accuracy page they also write:
Rounding in Monthly Printouts In monthly printouts, where seconds have
not been provided, all times for Sunrise, Earliest Talis, Earliest
Mincha and Nightfall (all opinions) have been rounded later.
Accordingly, a listing of "6:12" means "before 6:12" (between 6:11 and
6:12). All other Zmanim times have been rounded earlier. As such, a
listing of "5:36" means "after 5:36".
If you follow and read all the instructions on the website, you are left with the suggestion that the seconds on the zman are given so that you can know in which direction to round for the minute and to be cautious in it's accuracy.