docs: add Codex MCP setup and dev-proxy instructions

[debugmcpdev]

📋 Description

Adds first-party Codex setup instructions for the published mcp-debugger package and for source
development through the restartable dev proxy.

The README now includes the tested quick-start:

codex mcp add mcp-debugger -- npx -y @debugmcp/mcp-debugger stdio
codex mcp list

The dev-proxy guide now:

• uses client-neutral architecture and workflow wording;
• documents absolute proxy paths and DEV_PROXY_ROOT;
• covers Codex CLI, ChatGPT desktop, the Codex IDE extension, shared config.toml, and Claude Code;
• includes verification and client-restart guidance from the official Codex MCP documentation;
• corrects the documented fields returned by dev_server_status.

🔄 Type of Change

• 📚 Documentation update

✅ Checklist

• My code follows the style guidelines of this project
• I have performed a self-review of my own code
• I have made corresponding changes to the documentation
• My changes generate no new warnings
• New and existing relevant checks pass locally
• Any dependent changes have been merged and published in downstream modules (not applicable)

🧪 Testing

• Added the source dev-proxy entry with codex mcp add and verified it with codex mcp list.
• Completed a real MCP SDK handshake through that proxy: 31 tools and all nine languages reported.
• Completed a real MCP SDK handshake against the documented published npx ... stdio command:
  28 tools reported, beginning with create_debug_session.
• Ran pnpm check:all-personal-paths and git diff --check.

Test configuration: Windows NT 10.0.26200, Codex CLI 0.144.1, Node.js 24.14.1,
pnpm 10.33.0.

🔗 Related documentation

• https://developers.openai.com/codex/mcp

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.

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