First the output signal surpass the input signal (overshoot)
Initially the output signal follows the input exactly and goes up by the same amount relative to their current baselines. However the output signal quickly 'sags' back to the baseline as each capacitor in the filter charges, with a small undershoot caused by the second capacitor charging less due to the reducing output voltage from the first capacitor.
You could argue that this is 'overshoot' because an AC square wave goes above and below its true baseline and so has a lower amplitude, but in this time frame we (and the filter) cannot know that. It could just be a single step going from low to high voltage, and the filter has neither the 'clairvoyance' to see into the future nor sufficient 'memory' to account for what might have happened in the distant past.
What's confusing me is that this overshooting may be related to the
first harmonic (along with the higher frequency ones).
Frequencies below the cutoff frequency have less effect on the output because they are attenuated. What you are seeing is mostly the effect of higher frequencies, with the fundamental frequency determining the 'repetition rate'. Here are the results (simulated with LTspice) for fundamental frequencies of 1 Hz, 10 Hz, and 100 Hz:-
It makes virtually no difference to the impulse waveform whether the lowest harmonic is 0.1Hz, 1Hz, 10Hz or 100Hz. You could remove those lower frequencies entirely and the waveform wouldn't change much. It's not just some effect of being an 'active' op amp based filter either, a similar waveform is produced even with a passive RC filter:-