Flight numbers/call signs

Hi and apologies if this question has already been asked a million times!

Who actually assigns call signs to each individual flight ? Is it the airline or the aviation authority of the country the airline is from ?

It would cause a lot less confusion if they were the same. For instance. British Airways flight BA2159 LGW-UVF has the call sign BAW9VA, but you won`t find that flight on this website as it is shown as BAW2159.

The airline picks flight numbers for ticketing and operational use. To the extent possible we modify the ATC data to match the ticketing flight numbers so flights are easier to find.