[DOCS] Streamline felogin documentation

The patch provides the following changes:
- Use consistent header levels
- Format code examples with the correct language
- Use example.org as domain for example link
- Apply takeaways from https://forge.typo3.org/issues/85035#note-8
  (chapter "login process")

Resolves: #97692
Related: #85035
Releases: main, 11.5
Change-Id: If6c7851c25210cf4e03fb07efeaf14973e667cb9
Reviewed-on: https://review.typo3.org/c/Packages/TYPO3.CMS/+/74746


Tested-by: core-ci <typo3@b13.com>
Tested-by: Andreas Fernandez <a.fernandez@scripting-base.de>
Tested-by: Nikita Hovratov <nikita.h@live.de>
Reviewed-by: Simon Schaufelberger <simonschaufi+typo3@gmail.com>
Reviewed-by: Andreas Fernandez <a.fernandez@scripting-base.de>
Reviewed-by: Nikita Hovratov <nikita.h@live.de>