"Update available" banner shown even when on latest Homebrew version

Description

Claude Code shows an "update available" notification even when the user is running the latest version installed via Homebrew (brew upgrade claude-code confirms already up to date).

Environment

• Installation method: Homebrew cask (brew install claude-code)
• Current version: 2.1.81
• Platform: macOS (Darwin 25.3.0, Apple Silicon)

Steps to reproduce

1. Install Claude Code via Homebrew: brew install claude-code
2. Ensure it's up to date: brew update && brew upgrade claude-code → "already installed"
3. Launch claude — "update available" banner is displayed

Expected behavior

No "update available" banner when the user is on the latest version available through their installation method (Homebrew).

Actual behavior

"Update available" banner is shown despite being on the latest Homebrew version.

Notes

• The npm package @anthropic-ai/claude-code is deprecated (per the Claude Code GitHub page), so Homebrew is now the recommended install method on macOS.
• The update check likely still queries the npm registry, which creates a false positive for Homebrew users if versions are out of sync or if the check doesn't account for the Homebrew distribution channel.