[FEATURE] VSCode extension: replace the static Claude tab icon with a live session-status indicator

Preflight Checklist

• [x] I have searched existing requests — this re-raises #36089, auto-closed as stale and now locked (can't be reopened or commented on)
• [x] This is a single feature request (not multiple features)

Problem Statement

In the Claude Code VSCode extension, every open tab shows the same static Claude
icon regardless of what that session is doing. With multiple tabs open there's no
at-a-glance way to tell which session is busy, which finished, and which is paused
waiting for my input — I have to click into each tab to find out.

Proposed Solution

Replace (or augment) the constant Claude icon on each tab with an icon/badge that
reflects the session's current status:

• Running / thinking — animated spinner
• Waiting for user input — a distinct "needs attention" badge for when Claude

is paused on a permission prompt or question

• Done / idle — a checkmark or neutral icon when the last turn completed

Analogous to how VSCode already shows a spinner on tab titles for long-running
extension operations (language-server indexing, test runs).

Alternative Solutions

• Completion notifications only cover the "done" case and fire after the fact;

they don't show "still running" or "waiting for input" at a glance.

• Manually clicking each tab to check state — what I do today.

Priority

Medium - Would be very helpful

Feature Category

Other (IDE integration — VSCode extension UI)

Use Case Example

1. I start a long task in Tab 1, then switch to Tab 2 to start another.
2. Tab 1 shows a spinner (running), Tab 3 shows a "waiting for input" badge,

Tab 2 shows a checkmark (done).

1. I go straight to the tab that needs me instead of clicking through all of them.

Additional Context

Re-raises #36089 ("per-tab status indicator showing Claude running/idle state"),
auto-closed as stale (NOT_PLANNED) and now locked. Pairs naturally with per-tab
colors (#72209): color identifies which session, the status icon shows *what
state* it's in.