Why is the sky bright at night when it snows?

Surely I’m not the only one who has noticed this. When it snows at night, I’ve seen that the sky is really a bright grey, and you can see much better than when it’s not snowing. What’s up with that?

I think it’s because snow means there will be cloud cover, clouds reflect lights better than the clear sky.

Also when it is foggy or cloudy: The city lights are reflected back by the fog, snow etc. A clear sky does not reflect the light back.

Also, fresh snow reflects something like 88% of the light that hits it back into the air, where it can illuminate the underside of the clouds.