I think it depends on how badly they want to avoid it. Stuff that is "We prefer if the model didn't do these things when the model is used here" goes into the system prompt, meanwhile stuff that is "We really need to avoid this ever being in any outputs, regardless of when/where the model is used" goes into post-training.

So I'm guessing they want none of the model users (webui + API) to be able to do those things, rather than not being able to do that just in the webui. The changes mentioned in the submission is just for claude.ai AFAIK, not API users, so the "disordered eating" stuff will only be prevented when API users would prompt against it in their system prompts, but not required.

I wonder if the child safety section "leaks" behavior into other risky topics, like malware analysis. I see overlap in how the reports mention that once the safety has been tripped it becomes even more reluctant to work, which seems to match the instructions here for child safety.

It's built into the model, not part of the system prompt. You'll get the same refusals via the API.