docs: update landing auth example

[Kkkakania]

Closes #640.

Summary

• Update the landing page "Just Add #[public]" example to use the current CurrentUser extractor instead of the outdated Claims handler parameter.
• Adjust the surrounding copy and comment to match the current extractor docs.

Validation

• rg verified Claims / claims.sub no longer appear in docs/templates/index.html
• git diff --check passed

Note: I did not run the Zola docs build locally because zola is not installed in this environment.

[github-actions]

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[ShiraiEd]

Hi, great first contribution!

I have left 2 comments as suggestions that I think that would improve this a little more,
then once addressed, it will look got to merge to me

[Kkkakania]

Updated the branch to address both review suggestions:

• clarified the text to say the handler accesses the current user's data
• expanded the inline code comment to describe the authenticated user id lookup and JSON response

[ShiraiEd]

LGTM