Misleading 'Update available' message for Homebrew users

Description

When Claude Code is installed via Homebrew (brew install claude-code), the application displays an "Update available" message suggesting to run brew upgrade claude-code. However, running the command does nothing because the Homebrew cask is already at the latest available version.

Root Cause

It appears Claude Code checks the npm registry for the latest version (currently 2.1.112) and compares it against the installed version. Since Homebrew distributes a different (older) version (2.1.92), there is a version gap that the user cannot resolve via brew upgrade.

Steps to Reproduce

1. Install Claude Code via Homebrew: brew install claude-code
2. Confirm latest Homebrew version is installed: brew info claude-code → 2.1.92
3. Launch Claude Code → "Update available! Run: brew upgrade claude-code" message appears
4. Run brew upgrade claude-code → No update applied (already at latest brew version)

Expected Behavior

Either:

• Option A: The update check should compare against the latest Homebrew version, not the npm version, when installed via Homebrew.
• Option B: The message should inform the user that a newer version exists on npm but is not yet available via Homebrew, e.g., "A newer version (2.1.112) is available on npm but not yet on Homebrew. To get the latest, switch to npm: npm install -g @anthropic-ai/claude-code"

Environment

• OS: macOS (Darwin 25.4.0, Apple Silicon)
• Installation method: Homebrew cask
• Installed version: 2.1.92
• npm latest version: 2.1.112
• Homebrew latest version: 2.1.92