[FEATURE] Desktop: session status should reflect background task activity, plus more/customizable statuses & colors

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 Desktop app's Code tab, each session in the sidebar has a status dot (e.g.
"idle", shown gray). That dot only reflects the main agent's state — it does NOT
reflect background work. When subagents, background shell commands, or dynamic
workflows are actively running (visible only in the separate Tasks pane), the session
still shows "idle" because the main agent is waiting on them.

This is misleading: at a glance the session looks done/idle when it's actually busy. To
know whether anything is running I have to open the Tasks pane for that specific
session — there's no glanceable signal in the sidebar.

Separately, the set of statuses and their colors isn't customizable or documented, so
it's hard to tell states apart quickly when scanning many parallel sessions.

Proposed Solution

1. Add a distinct "working / busy" status (and dot color) that triggers when a

session has active background tasks — subagents, background commands, or workflows —
even if the main agent is idle. So "idle" means nothing is running, not "the main
agent is waiting."

1. Allow more granular and/or user-customizable status labels and dot colors

(e.g. idle / working / waiting-for-input / error / done) so session state is
glanceable in the sidebar.

1. (Nice to have) Document the full list of statuses and what each color means.

Alternative Solutions

Today the only way to know background work is running is to open the Tasks pane per
session (Views menu). There's no sidebar-level indicator and no way to customize or
even reference what the statuses/colors mean.

Priority

Medium - Would be very helpful

Feature Category

Other (Desktop app GUI)

Use Case Example

1. I dispatch a subagent / long background task in a session, then switch away.
2. The sidebar dot for that session shows "idle" (gray) because the main agent is

waiting on the subagent.

1. I assume nothing is happening and move on — but the task is still running.
2. With a "working" status + color, I'd see at a glance that the session is busy

without opening its Tasks pane.

Additional Context

Part of a small cluster of status-visibility requests: Desktop status line #71582 and
mobile status display #71583. This one is specifically about the sidebar session
status dot accurately reflecting background activity, plus customization.