An empty home and "~/.claude" resolve to the same directory but the CLI
treats an explicit CLAUDE_CONFIG_DIR as a separate login, so they must
not share probe results. Also stop caching failed probes so one
instance's failure is not served to its siblings for five minutes.

Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01CeWArRh6FDJGzjGYecyi7A