[FEATURE] Support user-level CLAUDE.md / custom instructions for claude.ai/code web sessions

Description

Claude Code CLI and desktop support a user-level ~/.claude/CLAUDE.md for personal conventions (branch naming, commit style, workflow preferences, etc.) that apply across all projects automatically. The web version (claude.ai/code) has no equivalent — only a project-level CLAUDE.md committed to a repo is loaded in cloud sessions.

Expected Behavior

Users can define personal instructions via a profile/settings page on claude.ai that are automatically applied to all Claude Code web sessions — similar to how ~/.claude/CLAUDE.md works for local sessions.

Actual Behavior

Web sessions only load CLAUDE.md from the repository. There is no user-level configuration surface on claude.ai for Claude Code personal instructions.

Use Case / Context

Personal conventions that shouldn't be committed to repos should still be usable in web sessions. Examples:

• Branch naming format: feat/vann/{task-description}
• Preferred commit message style
• Personal workflow preferences (e.g. always ask before pushing)

Currently, the only workaround is to either:

1. Commit conventions to every individual repo's CLAUDE.md (noisy, not always appropriate for shared repos), or
2. Use Remote Control to run Claude Code locally (defeats the purpose of the web version)

Proposed Solution

Add a "Personal Instructions" or "Custom Instructions" section under the claude.ai user profile/settings that Claude Code web sessions load at startup — analogous to ~/.claude/CLAUDE.md on local installs.