The human gait has a natural bobbing motion, with the head moving slightly up-and-down and side-to-side. The side-to-side motion (swinging on an axis parallel to the nose) turns the ponytail into a natural pendulum which swings back and forth, since this plane of motion is gravitationally symmetric and has nothing to stop the swing. Small driving forces can build up over time, causing a noticeable swing, very similar to how one would use a swing on a swingset.
The up-and-down motion (swinging on an axis parallel to the shoulders) does not turn the ponytail into a pendulum, because the hair cannot swing freely on this axis. The problem is, there no mechanism to conserve energy at the bottom of the up-down swing, since the ponytail hits the back of the runner's head and loses all its energy. For the side-to-side swing, there's a constant oscillation of gravitational potential and kinetic energy in the ponytail, which isn't so in an up-and-down swing - when the ponytail reaches the bottom of an up-down swing, it has lost all its potential and kinetic energy, so you can't keep imparting small forces which will grow over time and produce a repeating oscillation.
The front-to-back oscillations described in the question are the same as the up-and-down oscillations described in the previous paragraph (swinging along an axis parallel to the shoulders). The third axis of oscillation would be swinging on an axis parallel to the spine, which I think does happen to an extent. But since this axis is parallel to gravity, the ponytail hangs down very close to the axis, and rotations at this small radius tend to be lost in the much larger side-to-side swing. I suspect that the ponytail doesn't swing perfectly in a flat plane along only one axis, but actually wraps "around" the head slightly as it swings side-to-side - there may be a major swing along the axis of the nose, and a minor one along the axis of the spine.
In the end, the most noticeable swing is side-to-side along the axis of the nose. Up-and-down oscillations on the axis of the shoulders cannot build up over time with small driving forces. And since the ponytail hangs very close to the third axis of rotation (along the axis of the spine), these are of much smaller magnitude than the obvious swing along the axis of the nose.