Yeah, that's usually called "reasoning" or "thinking" tokens AFAIK, so I think the terminology is correct. But from the traces I've seen, they're usually in a sort of diary style and start with repeating the last user requests and tool results. They're not introducing new requirements out of the blue.

Also, they're usually bracketed by special tokens to distinguish them from "normal" output for both the model and the harness.

(They can get pretty weird, like in the "user said no but I think they meant yes" example from a few weeks ago. But I think that requires a few rounds of wrong conclusions and motivated reasoning before it can get to that point - and not at the beginning)