Boundaries between Approach/Departure and Center

A Guy Called Tyketto writes:

To answer your first question, there is no such thing as an
'approach or departure center'.

Well, there are frequencies that call themselves Approach or
Departure, and they are distinct from Center. For example, SoCal
Approach seems to cover an awful lot of ground, despite the "Approach"
in the name, and it's not clear which airport's approach it nominally
covers. At times I've wondered exactly what role Los Angeles Center
plays in the area, since it seems that Center never gets a handoff in
some regions.

The problem I have is somewhat unrealistic, because it is in VATSIM.
In real life, there are always controllers on ATC. In VATSIM,
sometimes there aren't (often, actually), and so when you enter an
area that does have a controller, you have to establish contact with
ATC. The part that mystified me is that sometimes Center wants you to
contact them, and sometimes Approach or Departure Control. I suppose
in real life you'd be continuously in contact with ATC, and they would
handle all handoffs, and so you wouldn't have to ask yourself whom to
contact.

However, on charts there are still indications of Approach
frequencies, so if you are not in contact with ATC you can get into
contact with them as required. But the sectionals show only little
boxes, not the limits of ATC coverage. I know I have to establish
communication with ATC in some circumstances (entering Class C, for
example); are these boxes intended to tell me whom to contact when I
do establish contact, or what? There's already a separate list of
frequencies in the legend of the chart, plus the A/FD, so I'm left
scratching my head at times.

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