[FEATURE] VS Code extension: /rename should update tab title

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

Feature Request
When using Claude Code in the VS Code extension, the tab title is set to the first prompt and cannot be changed.

Current Behavior

• Tab title is automatically set from the first message in the conversation
• /rename <name> updates the session name (visible in session picker / /resume)
• Tab title remains unchanged after /rename

Environment

• VS Code extension (not terminal mode)
• macOS

Proposed Solution

Expected Behavior
/rename <name> should also update the VS Code tab title, or there should be another mechanism to change the tab title.

Alternative Solutions

_No response_

Priority

Medium - Would be very helpful

Feature Category

CLI commands and flags

Use Case Example

Use Case
When working on multiple sessions, the tab titles become confusing since they all show truncated versions of the first prompt. Being able to rename tabs to meaningful names (e.g., "refactor-auth", "marketing-docs") would significantly improve the UX when managing multiple conversations. Also, session use changes with time and moves on to other topics.

Additional Context

_No response_