[FEATURE] Claude Code should respect which Homebrew cask the user has installed when checking for updates

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

The Claude Code app occasionally notifies the user that a new version is available.
However, this notification happens no matter which Homebrew cask the user has installed.
This leads to 'false' notifications if the new version is not present in the installed cask.

Proposed Solution

Have the Claude Code app check to see which cask the user has installed (claude-code or claude-code@latest) and check that cask for updates. For example, if Claude Code ran the 'brew list' command, it could know which cask the user has installed, and therefore know which cask to check for updates.

Alternative Solutions

Ignore the 'new version available' message, and only check for updates using 'brew update ; brew outdated', since that will only check the installed cask.

Priority

Medium - Would be very helpful

Feature Category

Other

Use Case Example

Example scenario:

1. I am using the Claude Code app
2. The Claude Code app notifies me that 'a new version is available'
3. I can trust that this notification applies to my Claude installation (i.e. that it refers to the cask I have installed)

Additional Context

_No response_