Pulse distortion low pass filter and group delay
I would call a pulse distorted if its shape (value over time) and/or its spectrum (frequency content) has changed between input and output of the filter.

Theoretically an ideal pulse (with infinitely steep slopes) has an infinite spectrum, that means that anything that changes that spectrum will distort the pulse. A filter will change the spectrum so it will distort the pulse.

Even if the filter would have a constant group delay (a constant delay for all frequencies) but the filter would still affect the amplitudes of some frequencies the pulse would distort as the spectrum would be changed.

If the filter would have a constant group delay (a constant delay for all frequencies) and not affect the amplitude of any frequency the pulse would not distort, it would only be delayed (in time).

If your input pulse is not ideal, meaning its slopes are limited in their steepness, then the spectrum of this pulse will not extend into infinitely high frequencies. Instead it will have a limited spectrum.

If the filter only affects (attenuates) the frequencies that are not present in the spectrum of the pulse and the filter has a constant group delay for all frequencies, then the pulse would not be distorted as nothing was changed by the filter.

So: it depends on the pulse and the filter. With an ideal pulse, filtering it will change (distort) the pulse.