Expand and adapt usage examples

This MR adds usage examples to the docs, integrating them from custom tests.

Anything to double-check?

Are the examples.rst and changes to philosophy.rst understandable like this?

Can this MR be accepted?

• Add test_doc_examples test module
• Extended documentation
  • Added examples.rst (test-based)
  • Added (empty) integrating.rst
  • Extended philosophy.rst
  • Migrated some other code snippets to be test-based
• Pipeline passes without warnings
• Changelog entry added
• Version number bumped
• History cleaned up
• Approved by @herdeanu and @jeremiastraub

Related issues

Meta task: #117 (closed)