Two counting conventions, one of them on a switch
The number of days between 1 January and 31 December is 364, or 365 if the first day counts as well. Neither is wrong: a timesheet counts both ends of a shift, a hotel counts the nights. The switch says which convention the headline number uses, and the other number is printed underneath so nobody has to guess.
Weekdays and weekend days are counted under the same convention, so they always add up to the total. Friday to Monday with the end date counted is four days: two weekdays and two weekend days. The same pair without the end date is three days: one weekday and two weekend days.
Public holidays are deliberately not subtracted. There is no single list of them - they differ by country, by state, by year, and by whether a substitute day was granted - so a page that guessed would be wrong on precisely the day it mattered. If you need working days rather than weekdays, take the weekday count and subtract your own holiday list.