Self-distillation generally refers to training a smaller model, right? I suppose for full metacognition you would use it fine-tune the existing model based on its older self?
"Self-distillation" refers to distilling from a model into a copy of itself. Which is of limited use - unless you can steer the teacher, and want the student to internalize that steering.
The reason for doing self-distillation here is that we have both access to a richer representation (logit stream), and want to capture a richer behavior - not the answers themselves, but better reasoning techniques that are downstream from better prompts.