Rename and move duo-chat example configs

What does this merge request do and why?

Move and rename configs, plus minor tweaks.

Follow up on !479 (comment 1956986612)

Resolves #328

How to set up and validate locally

Numbered steps to set up and validate the change are strongly suggested.

Merge request checklist

• I've ran the affected pipeline(s) to validate that nothing is broken.
• Tests added for new functionality. If not, please raise an issue to follow up.
• Documentation added/updated, if needed.