[FEATURE] VS Code: tab icon indicator for sessions that are currently working

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

In the VS Code extension I usually keep several Claude Code tabs open, each on a different task. When I switch away from a tab there is no way to tell, from the tab strip alone, which sessions are still running and which are idle.

The tab icon today has only three states (I checked applyTabIcon in the extension bundle, v2.1.292):

| state | icon | when |
|---|---|---|
| pending | blue dot | permission request waiting |
| done | orange dot | finished while the tab was hidden |
| plain | no dot | everything else, including "currently working" |

The function that picks the state already knows tab.state === "running", but only uses it to suppress the orange dot. So an idle tab and a tab that has been running for ten minutes look identical. To find out which tabs are busy I have to click through each one, or open the sessions list in the Activity Bar and apply the Active filter.

Proposed Solution

Add a fourth tab-icon state for a session that is actively running a turn, e.g. a working state with its own icon (a dot in a distinct color, or the same spinner the webview already shows next to the prompt box). Priority order would be:

1. pending (blue) – needs my input, highest priority
2. working (new) – running, nothing to do yet
3. done (orange) – finished while hidden
4. plain

This mirrors what the Activity Bar sessions list already exposes as Working / Needs input / Completed, but puts it where the eye already is when juggling tabs.

A setting to turn it off (claudeCode.showWorkingTabIndicator or similar) would be nice but not required.

Alternative Solutions

• The Active filter in the Activity Bar sessions list works but requires opening a side panel.
• A Stop hook with an OS notification tells me when a session finishes, not which ones are still going.
• Patching the extension bundle to add a working entry to the icon map works, but is overwritten on every extension update.

Priority

Medium - Would improve my workflow

Additional Context

• VS Code extension v2.1.292, macOS (Darwin 25.6.0), native UI (not useTerminal).
• Relevant code: function do1($,J,Q) → returns "pending" | "done" | "plain"; io1 = {pending:"claude-logo-pending.svg", done:"claude-logo-done.svg", plain:"claude-logo.svg"}.