getting arbitrary information by resolving to write it down once she knows it, then seeing herself write it down, seems different

Specifically, she'd have to actually have a path forward to learning the information, within the horizon that she can see, outside of any "shadows" cast by wolves or hybrids, that isn't influenced by too many other people's shaky choices. This means she'd have to be reasonably likely to learn the information (and not be misled by misinformation) if she didn't make the prediction. If Bella commits to informing Alice of something in the future conditional on Alice not yet knowing that thing, that's a clear path to the information.