Polish menu and overlay states for first-time users

[rain2day]

Goal

Improve the first-run experience when usage data or the Codex pet is not visible yet.

Ideas

• Make empty/missing-data states clearer from the menu.
• Add a menu action that opens the diagnostic log path or copies it safely.
• Review labels for Codex-only, Claude-only, and combined modes.
• Keep the overlay lightweight and avoid explanatory clutter.

Done When

• A new user can tell whether the issue is missing pet state, missing credentials, provider backoff, or app not running.
• No raw credentials are shown in UI or logs.