I still don't understand what the value of specifying a license for generated images is - at least in terms of enforceability. How could anyone reliably, conclusively determine that an image was generated using a locally-run tool?
I suppose in the case of DALL-E they probably save a copy of every generated image and can use some sort of reverse image search and find if/when the image was created.
With StableDiffusion or any other local tool I don't see how that would be possible. The best I think one could do is come up with a secondary tool that can pinpoint the position in latent space a generated image came from (if that's even possible, I have a limited understanding of how exactly these systems work.) But then, if (heaven forbid) someone applied any sort of cropping or post-processing to the image, an approach like that easily gets blown out of the water.
That is not so; the CC0 explicitly states that patent and trademark rights are not waived.
Contrast that with, say, the Two-Clause BSD which says "[r]edistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met [...]".
Since trademark and patent rights are not mentioned, then these words mean that even if the purveyor of the software holds patents and/or trademarks, your redistribution and use are permitted. I.e. it appears that a patent and trademark grant is implied if a patent holder puts something under the two-clause BSD. Or at least you have a ghost of a chance to argue it in court.
Not so with the CC0, which spells out that you don't have permission to use any patents and trademarks in the work.