[FEATURE] Desktop: show where a session runs (local / SSH / Remote Control) in the session list

Problem. I run Claude Code sessions in several places: on my laptop (Local), on my home server over SSH from the Desktop app, and on the same server through Remote Control from claude.ai and the mobile app. In the Desktop sidebar all of these look the same. The only way to tell is to open each session. With 20+ sessions a day this is the main source of confusion, and it's easy to keep working on the wrong machine.

What I tried.

• Grouping by project/folder works, but then I lose the date grouping (Today / Yesterday), which I need more.
• A mod can't draw in the sidebar. A mod running on the server can /rename the session (we prefix "🧠"), and that shows up for Remote Control sessions, but SSH sessions from the Desktop app keep the app's own title.
• Custom groups only hold local sessions.

Request.

1. A small icon or badge per session in the sidebar for its environment: Local, SSH host name, Remote Control machine, Cloud.
2. Optionally, a filter or a second grouping level ("by date, within environment"), or a way for a mod to add a badge to a session row.
3. Bonus: when an SSH session is renamed with /rename on the remote host, update the Desktop sidebar title too.

Environment. Claude Desktop on Windows 11, Claude Code 2.1.286 (Desktop) and 2.1.289 (Linux server), Max plan.