[FEATURE] Show a working/activity indicator in the Claude Desktop Chat tab while a response is generating (like the Code tab)

Preflight

• I searched the existing enhancement requests and did not find this already reported.
• This is a single feature request.

Problem Statement

In the Claude Desktop app, the Chat tab gives no clear visual indication that it is actively working while a response is being generated. The Code tab already shows a working/activity indicator, so you can tell at a glance that it is busy.

The Chat tab has no equivalent. When a chat is "running" (waiting for the first token, processing a longer request, etc.), it is hard to tell whether the app is genuinely working or has stalled/frozen.

Proposed Solution

Add a clearly visible working/activity indicator to the Chat tab that appears whenever a response is in progress — matching the affordance the Code tab already provides. For example, a spinner or an animated "working…" state near the message/input area that stays visible until the response completes.

Alternative Solutions

Today the only cue is streamed text appearing, which is ambiguous during pauses (before the first token, or during longer processing). Reusing the same indicator pattern the Code tab already uses would keep the two tabs consistent and require no new visual language.

Priority

Medium — would be very helpful.

Additional Context

This is purely about the Desktop app's Chat tab UI. Bringing it in line with the Code tab would make the Chat tab feel more responsive and remove the "is it working or is it frozen?" uncertainty during longer responses.