docs(claude): update multi-account setup for CLAUDE_CONFIG_DIR

[markoapple]

What changed

• Updated the Claude provider guide to use CLAUDE_CONFIG_DIR instead of overriding HOME
• Updated the provider field names and multi-account examples
• Added a note for users who followed the older HOME-based instructions

Why

T3 Code now passes custom Claude paths through CLAUDE_CONFIG_DIR, so the existing instructions no longer match the app and can leave previously configured accounts appearing logged out.

Fixes #4696

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Note

Low Risk
Documentation-only changes with no runtime or security impact.

Overview
Updates the Claude provider guide so it matches how T3 Code isolates multi-account and router setups today.

Settings and terminology now refer to CLAUDE_CONFIG_DIR path instead of Claude HOME path, and prose describes separate configuration directories rather than overriding HOME.

Setup examples use CLAUDE_CONFIG_DIR=... claude auth login and shorter example paths (e.g. ~/.claude_personal, ~/.claude_openrouter, ~/.claude_router) instead of HOME=... and *_home directories.

Adds a migration note for users who followed the old HOME instructions: log in again with CLAUDE_CONFIG_DIR set to the path configured in T3 Code.

Minor doc cleanup: bullet lists use * consistently, and external links are plain URLs instead of angle-bracket markdown links.

^{Reviewed by Cursor Bugbot for commit 4818abf. Bugbot is set up for automated code reviews on this repo. Configure here.}

Note

Update Claude multi-account setup docs to use CLAUDE_CONFIG_DIR instead of HOME

Updates claude.md to standardize multi-account setup guidance around CLAUDE_CONFIG_DIR rather than overriding HOME.

• Replaces all references to 'Claude HOME path' with 'CLAUDE_CONFIG_DIR path'
• Updates login commands from HOME=... claude auth login to CLAUDE_CONFIG_DIR=... claude auth login
• Shortens example config directory names (e.g. ~/.claude_personal_home → ~/.claude_personal)
• Converts hyphen bullet lists to asterisk bullets and cleans up URL formatting

^{Macroscope summarized 4818abf.}

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[macroscopeapp]

Approvability

Verdict: Approved 4818abf

Documentation-only changes updating the multi-account setup guide to use CLAUDE_CONFIG_DIR instead of HOME. No code or runtime behavior is affected.

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