The confusion may come from that fact that if you
are receiving flight following (or IFR) from center or
some other facility and you fly through the class C
you are ok, it is there responsibility to coordinate
with the class C controllers for you or to hand you off.

IFR, yes, but flight following, no. That's the whole point of the FAA's
reply. If you're getting flight following, it's still your
responsbility to know where you are and to make sure you're talking to
the TRACON before you enter the Class C. If Center doesn't hand you
off, that doesn't absolve you.