> Once you correct the LLM, it will continue to provide the corrected answer until some time later,

That is only true if you stay within the same chat. It is not true across chats. Context caching is something that a lot of folks would really really like to see.

And jumping to a new chat is one of the core points of the OP: "I restarted with a slightly modified prompt:"

The iterations before where mostly to figure out why the initial prompt went wrong. And AFAICT there's a good insight in the modified prompt - "Make no assumptions". Probably also "ensure you fully understand how it's labelled".

And no, asking repeatedly doesn't necessarily give different answers, not even with "can you double check". There are quite a few examples where LLMs are consistently and proudly wrong. Don't use LLMs if 100% accuracy matters.