gh-36919: refresh the doc about coercion and test the given example
There is one example of ring in the doc about coercion.
Currently not tested and very much using old-style.
So let's refresh it and test it.
### :memo: Checklist
- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation accordingly.
URL: https://github.com/sagemath/sage/pull/36919
Reported by: Frédéric Chapoton
Reviewer(s): Ruchit Jagodara
