QUESTION:
Why it is that 2:09 a.m. on 2 December (and earlier times on dates
thereafter) is the precise time Pathfinder must launch? What is is about the
Earth's position that makes this important? Since the Pathfinder is put in
orbit over Earth anyway, it would seem that the only important thing is when
you turn on the leftover of stage II and stage III in order to get it in the
direction of Mars at the right time.

ANSWER from David Dubov:
When the spacecraft leaves the Earth to go to Mars, it must be going in a
particular direction. Since the Earth rotates, the launch site is only lined
up with this direction twice per day (for an instant in each case). Since
the two opportunities are about 12 hours apart, the launch vehicle people
make us choose one or the other. It is okay to launch at a time slightly
different from the ideal time because the spacecraft can use it's propulsion
system to correct for the error. The spacecraft has a limited amount of
fuel, however, so we can't accept a very big error (up to approximately 1
minute is okay).

You can see all of Mars Pathfinder's launch opportunities at the Launch
Windows Page. (http://mpfwww.arc.nasa.gov/mpf/launch_windows.html)

ANSWER from David Dubov:
You're correct in saying that once we're in Earth orbit, it is necessary to
time the stage II/stage III burns properly to inject us
onto the interplanetary transfer trajectory. It is also critical, however,
that the circular parking orbit be in the same plane as the
interplanetary trajectory. This allows us to take advantage of the velocity
our spacecraft has from the parking orbit; the launch
vehicle does not have enough energy to drastically change our orbit plane.
Since the plane of the parking orbit is critical, the
launch time is also critical. Think of it this way: for a "planar" launch,
the launch site, the center of the Earth, and the
interplanetary velocity vector must all lie in the same plane. This only
happens twice each day.