No visual indicator distinguishing cloud sessions from local sessions in the session list (desktop)

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

When I switch back to my desktop after starting a Claude Code session from another device (mobile), the session list shows no visual distinction between sessions running in a cloud container and sessions running locally.

I was perfectly aware that I started a cloud session from my phone — that's not the issue. The confusion happens on desktop, when looking at the session list: all sessions look identical regardless of their execution environment. The session name can sometimes hint at the context, but there is no reliable, at-a-glance indicator.

Proposed Solution

A simple visual cue in the session list would be enough — for example:

• A color difference
• A cloud// prefix in the session name
• A small badge or icon

Alternative Solutions

_No response_

Priority

Low - Nice to have

Feature Category

Interactive mode (TUI)

Use Case Example

1. I start a Claude Code session from my phone (cloud environment is created)
2. I switch back to my desktop and open Claude Code
3. I see a list of sessions — cloud and local ones look identical
4. I ask Claude to run git pull on my local repo, not realizing the session is running in a cloud container that has no access to my local filesystem
5. A simple cloud// prefix or badge on the session would have made the context immediately clear

Additional Context

The execution environment determines what Claude can and cannot do — for example, a cloud session has no access to the local filesystem. Without a clear indicator, it's easy to try delegating local tasks (like git pull on a local repo) to a session that simply cannot perform them. This is especially confusing for users who are not already familiar with remote execution concepts.