Add custom session label field in extension header

Problem

When running multiple Claude Code sessions in parallel (different worktrees, different tasks throughout a working day), the session tabs in the VSCode extension show auto-generated titles derived from the current conversation. These auto-titles:

• Change as the conversation evolves
• Can be identical or near-identical between sessions on similar topics
• Don't reflect why I opened that session or which task it belongs to
• Cannot be renamed via the standard VSCode tab right-click menu — no "Rename" option appears for Claude Code tabs (only Close, Pin, Split, etc.)

This makes context-switching between sessions very confusing, especially when juggling several worktrees and parallel tasks all day. Pure dev ergonomics issue.

Proposed solution

Add either:

1. A small editable text field in the session header where the user types a short label, OR
2. A "Rename session" option in the tab context menu

The label should:

• Persist for the lifetime of that session
• Be shown in the tab title (prefixed or replacing the auto-title)
• Be a pure UI affordance — does not affect the conversation, prompt, or model behavior

Example use case

I run 3 parallel worktrees on the same repo, each with its own Claude Code session. I'd like to label them:

• 236 chat DM
• 239 schedule wip
• 258 jitsi POC

Instead of guessing from auto-titles which session belongs to which worktree.

Workarounds tried

• VSCode tab right-click → no Rename option appears for Claude Code tabs
• Relying on the auto-title → drifts as conversation evolves
• Reading the working directory in the header → present but easy to miss when many tabs are open
• Custom terminal tab rename → doesn't apply, these are extension panels not terminals

<img width="852" height="985" alt="Image" src="https://github.com/user-attachments/assets/e5ff3a11-6b55-4f13-bd65-70378df3a516" />

Environment

• VSCode native Claude Code extension
• Linux