Add public database interface to MistDemoApp and web

[leogdion]

Summary

Add public database support to both the MistDemoApp (SwiftUI) and the MistDemo web interface.

Context

From PR #273 review feedback: currently both the native app and CLI default to the private database only. Users should be able to switch between public and private databases.

Scope

• MistDemoApp (SwiftUI): Add a database picker (public/private) and wire NativeCloudKitService to use the selected database
• MistDemo web: Once web CRUD routes exist, support public database operations
• Public database uses server-to-server auth (CLOUDKIT_KEY_ID + CLOUDKIT_PRIVATE_KEY), private uses web auth — the UI should reflect which credentials are needed

🤖 Generated with Claude Code

[leogdion]

Resolved by #337 (commit bad7b1e, web side; native side covered by #339).