The BBC had just made a compete nonsense of reporting this, or the LGA itself doesn't understand the system.
a) HT decides whether or not to authorise (can authorise, for a number of reasons, including holidays of any length in exceptional circumstances)
b) LEA sets fines policy, including whether they use them at all.
c) Schools must report those with unauthorised absences, and if they pass who pass the LEA's trigger point, then LEA fine them.
d) excessive absences (all reasons) are the responsibility of the EWO (an LEA function), who is responsible for deciding on support/sanctions for the family.
The HT does not fine the family, and I'm sure I've read on MN that there is at least one LEA which does not impose fines at all.
So either LEAs don't get this (a system unchanged since 2003 so a shocking lapse), or LGA doesn't get it (in which case it's a major mistake to put out a press release), or BBC is being remarkably crap to the point of being misleading.