Why is distorted / quantized white noise still white noise? White noise can be created by setting every sample in the time domain to a random number with a gaussian distribution. Every sample is random, so the frequency response is obviously random. Easy enough.

Next step: Set every sample to a random number between 0 and 1, but the numbers are equally distributed instead of using a gaussian distribution (please ignore the DC offsets, they don't really matter). Samples are random enough, so it still makes sense that it would sound like white noise.

Now, set every sample to either 0 or 1 with a 50/50 chance. This feels like it shouldn't sound like white noise anymore, but it still does.

Finally, set every sample to either 0 or 1 with a 90/10 chance. How does this still sound like white noise when it looks nothing like it?

EDIT
I did some experimenting (with sample rate 48000 Hz).
1/2 chance is basically white noise.
1/4 chance sounds almost like white noise, but if you know what you're looking for it's easy to tell the difference.
1/16 chance sounds like rain smattering against something (you can hear individual impulses), with white noise in the background.

Varying the amplitudes of the impulses did not seem to have any effect on the sound, other than volume.
White noise is defined as having a flat spectrum over the entire frequency range. Using uncorellated random numbers is just one of many methods to generate this, but by far not the only or the best one.
Also, whiteness implies no gaussian or whatever distribution, so your first sentence is wrong twice.