server: Update and expand READMEs

[s-heppner]

Previously, we adapted our server project structure and with that, split our project-level README.md into smaller parts, in preparation for the separate ./server/example_configurations.

Now we should make sure that the README.md files are up to date and expand them were necessary with new information on how to run it.

[ArokyaMatthew]

@s-heppner , I would like to contribute on this issue.

[s-heppner]

Thanks for the interest!

This issue is mainly an internal reminder related to a recent refactoring of the server. Updating the README.md files requires context about the design decisions behind those changes, which isn't fully captured in the issue itself.

Because of that, we’ll keep this issue with the maintainers for now. Feel free to take a look at other open issues if you’d like to contribute, for example #460, or #198.