[FEATURE] Add `installMethod` support for system package manager installations (pacman/AUR, apt, etc.)

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

Summary

Claude Code installed via pacman on cachyos (and I assume other arch-based distros) has no valid installMethod value, causing persistent startup warnings.

Current Behavior

When installed via pacman (Arch Linux AUR package), ~/.claude.json defaults to "installMethod": "unknown", producing:

Warning: Native installation exists but ~/.local/bin is not in your PATH
Warning: Running native installation but config install method is 'unknown'
Fix: Run claude install to update configuration

Problem

This warning is an annoyance and claude code is working fine and being updated diligently so it's not necessary.

Environment

• Claude Code version: 2.0.60
• Platform: CachyOS Linux (pacman/AUR)
• Binary location: /usr/bin/claude (symlink from /opt/claude-code/bin/claude)

Proposed Solution

Add a new valid value like "system-package" or "package-manager" that:

1. Suppresses the misleading warnings
2. Accurately reflects the installation method
3. Skips self-update prompts (since updates come from the package manager). I think mac/homebrew does this because the warning on macs says to run brew update.

Alternative Solutions

_No response_

Priority

Low - Nice to have

Feature Category

Configuration and settings

Use Case Example

_No response_

Additional Context

_No response_