docs: reorganize files to match nav sections, add ADK + comparison updates

[zeroasterisk]

Summary

Reorganizes documentation files to match their navigation sections, updates content for React renderer launch, improves community contribution guidance, and adds the web_core shared library section to the renderer development guide.

File Moves

File	From	To
catalogs.md	docs/	docs/concepts/
transports.md	docs/	docs/concepts/
client_to_server_actions.md	docs/	docs/concepts/
renderers.md	docs/	docs/reference/
agents.md	docs/	docs/reference/
community.md	docs/	docs/ecosystem/
where-is-it-used.md	docs/introduction/	docs/ecosystem/a2ui-in-the-world.md

All old URLs redirect via mkdocs-redirects plugin.

Content Updates

React renderer merged (#542)

• Updated renderer status from "In Review" to ✅ Stable across all pages
• Added React section to client-setup guide with install instructions
• Added React to "How Can I Use It?" quick setup
• Linked to React shell sample

Renderer development guide

• New section: "Web Renderers: Use web_core" — explains the shared library that Lit, Angular, and React all build on, with import examples and version support table

Community & ecosystem

• Expanded "Submitting a Renderer" with step-by-step contribution guide
• Replaced "Coming soon..." placeholder with concrete guidance and sample links
• Collapsed duplicate partner info (community.md now points to a2ui-in-the-world.md)
• Added live shields.io badges (⭐ stars, last commit) to community renderers table
• Fixed ChatKit link to actual docs page
• Fixed typo: "signiciant" → "significant"

Comparison & positioning

• Rewrote "How Does A2UI Compare?" as concise comparison with table
• Added Google ADK section to "A2UI in the World"
• Credited CopilotKit for Composer (not React renderer)

Links & tables

• All renderer names in tables now link to source code
• Fixed broken relative paths between ecosystem docs
• Added redirects for all moved files

No structural nav changes

The nav section names and ordering are unchanged. Only file paths were updated to match.

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[gemini-code-assist]

Code Review

This pull request effectively reorganizes the documentation by moving files into new directories that align with the navigation structure. It also includes significant content updates, such as rewriting the "How Does A2UI Compare?" page and adding a new section for the Google Agent Development Kit (ADK). All file moves are accompanied by corresponding link updates and redirects in mkdocs.yaml to prevent broken links. My review identified a couple of links in the documentation that could be improved for accuracy and specificity. Overall, this is a great improvement to the documentation's structure and content.