That's a strange one... I would actually call their definition of "Gaussian frequency response" a misnomer too. ("A low-pass frequency response that has a slow roll-off characteristic that begins at approximately 1/3 the –3 dB frequency".)
Every other text I have seen defines a "Gaussian frequency response" very specifically as a frequency response that follows Gauss' "bell curve", i.e. g(f) ~ exp (-f²/a). This specific frequency response is not required to obtain the 0.35/risetime relationship, and it is not present in typical scope front ends.
All actual derivations of the 0.35 factor which I have come across assume a plain old first-order or second-order lowpass filter. Probably a real Gaussian filter would give a similar factor. But focusing the whole bandwidth discussion on Gaussian filters, as Keysight does, seems misleading to me.
Primarily, a Gaussian filter has a Gaussian impuse response, and that results in a lowpass filter (not bandpass as you said in a previous post). The magnitude of the frequency response also has a Gaussian bell shape, centered at 0Hz (if you consider both, positive and negative frequencies in the Fourier frequency domain). And the corresponding log magnitude (-> decibel scale) is a quadratic function of frequency.
[ Generally, bandpass filters can be derived from their corresponding lowpass prototypes. So Gaussian bandpass filters do exist as well. However, a Gaussian bandpass does not have a Gaussian impulse response (it's a cosine wave with Gaussian envelope). ]
The risetime * bandwidth product of an
ideal Gaussian (lowpass) filter is 0.332.
A filter with Gaussian impulse response has the shortest rise time which is possible
without overshoot in the step response. There exist filters with same bandwidth and shorter rise times, but they do overshoot. A "maximally flat" frequency response will always overshoot.
A true Gaussian response can only be approximated, in both domains, analog and digital. At least in the analog domain, never expect a perfect approximation. In the digital domain, the approximation accuracy is, of course, just a matter of the number of FIR filter taps you are willing to spend.