alt Hacker NewsIMO it depends a bit, but in most cases: No!
If you do proper software development (planing, spec, task breakdown, test case spec, implementation, unit test, acceptance test, ...) implementation is just a single step and the generated artifact is the source code. And that's what needs to be checked in. All the other artifacts are usually stored elsewhere.
If you do spec and planing with AI, you should also commit the outcome and maybe also the prompt and session (like a meeting note on a spec meeting). But it's a different artifact then.
But if you skip all the steps and put your idea directly to an coding agent in the hope that the result is a final, tested and production ready software, you should absolutely commit the whole chat session (or at least make the AI create a summary of it).
LLMs frequently hallucinate and go off on wild goose chases. It's admittedly gotten a lot better, but it still happens.
From that perspective alone the session would be important meta information that could be used to determine the rationale of a commit - right from the intent (prompt) to what the harness (Claude code etc) made of it. So there is more value in keeping it even in your second scenario