[FEATURE] VSCode extension: per-tab status indicator showing Claude running/idle state

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

When using the Claude Code VSCode extension with multiple tabs open, there is no at-a-glance indicator of whether Claude is actively processing a request or idle in any given tab.

Current behavior:

• You can have multiple Claude Code chat tabs open simultaneously
• When you kick off a task in one tab and switch to another tab, there is no visible indicator that the first tab is still running
• A VSCode notification fires only when the task is finished — you only learn the result after the fact
• To check if Claude is still running in a tab, you have to manually click into that tab and look at the chat

Why this matters:

• Multi-tab workflows are common: start a long task in Tab 1, switch to Tab 2 to work on something else, Tab 3 for another context, etc.
• There is no ambient awareness of which tabs are active/processing vs. idle
• You waste time either switching back to check status, or miss the moment a task finishes and you can give the next input

Proposed Solution

Add a small status indicator to each Claude Code tab in the VSCode panel that reflects the current state:

• Running / thinking — a spinner or animated icon on the tab title (e.g., ⟳ Claude or ● Claude)
• Finished / idle — a checkmark or badge when the last request completed (e.g., ✓ Claude)
• Waiting for input — an indicator when Claude is paused and waiting for user permission or response

This is analogous to how VSCode shows a spinner on the tab title for ongoing operations in other extensions (e.g., language server indexing, test runners).

Implementation options:

1. Modify the tab/panel title dynamically (e.g., prefix with a Unicode spinner or status char)
2. Add a small colored dot/badge to the tab using VSCode's TreeItem.iconPath or webview title API
3. Update the activity bar icon badge count (similar to how source control shows pending changes)

Alternative Solutions

The existing VSCode notification on completion partially solves the "done" case, but it does not solve the "is it still running?" question. A tab-level indicator addresses both states without requiring the user to navigate away from their current work.

Priority

Medium - Improves my workflow

Feature Category

IDE integration (VSCode extension)

Use Case Example

1. I open three Claude Code tabs: one for refactoring a module, one for writing tests, one for documentation
2. I start a long task in Tab 1, then switch to Tab 2 to start another task
3. At a glance, I can see Tab 1 has a spinner (still running), Tab 2 just finished (checkmark), Tab 3 is idle
4. I know exactly which tabs need my attention without manually clicking into each one

Additional Context

Related but distinct from #36085 (cross-platform system notifications for Apple Watch etc.) — this is specifically about in-editor, per-tab visual state within the VSCode extension UI, not OS-level notifications.