There's still a lot of UV that comes in through the cloud layer. It's one of the reasons it's recommended to wear sunglasses and sunscreen if you're hiking on snow on a cloudy day. You can still sunburn the inside of your eyes from UV exposure, and because it's not visible light your iris doesn't react to it the same way it would on a bright sunny day.

But, as a cause of skin cancer, that would preclude vitamin D deficiency, since it is also dependent on UV exposure.

UV is a spectrum. UV-B is responsible for vitamin D production. Very little UV-B reaches us, even during the summer in the northern hemisphere. UV-A does damage, and reaches us much more strongly. You can receive damaging UV exposure while producing little to no vitamin D.

https://en.wikipedia.org/wiki/Ultraviolet#/media/File:Ozone_...

I see.