Upgrade notification shown to Homebrew users who are already on latest cask version

Description

When Claude Code is installed via the claude-code Homebrew cask, the upgrade notification incorrectly tells users an upgrade is available even when they are on the latest Homebrew version.

Root cause

The version check compares the installed version against the latest version published to npm (@anthropic-ai/claude-code), but the Homebrew cask typically lags behind npm by a short period. This means Homebrew users see an upgrade prompt they cannot act on.

Steps to reproduce

1. Install Claude Code via Homebrew: brew install claude-code
2. Upgrade to the latest available cask version: brew upgrade claude-code
3. Launch claude — an upgrade notification is displayed despite being on the latest Homebrew version

Example

Installed (via brew cask): 2.1.33
npm latest:               2.1.34
Homebrew cask latest:     2.1.33

The notification suggests upgrading, but brew upgrade claude-code reports already at the latest version.

Suggested fix

The version check could either:

• Detect the installation method and compare against the appropriate source (npm vs Homebrew)
• Adjust the upgrade instructions based on install method (e.g., suggest brew upgrade claude-code instead of npm update)
• Suppress the notification when the delta is very recent / minor and the install source differs

Environment

• macOS (Darwin 25.2.0)
• Installed via: brew install claude-code
• Claude Code version: 2.1.33