[FEATURE] Show which machine a Remote Control session is running on

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 open a Remote Control session from another device (laptop or phone), the Claude desktop app shows the same "computer" icon next to the session title as on the host machine. There is no visible indication of which machine actually executes the session.

I used to run a project on my laptop and moved it to my office desktop; both machines have a copy of the project folder. When viewing from the laptop, I cannot tell whether my commands will run on the desktop (Remote Control) or on the laptop's local copy (a new local session). For a live trading bot this matters: running in the wrong folder could start stale code.

Proposed Solution

• Show the host machine name (e.g. "Running on SIMON2022") and a Remote Control badge in the session header and sidebar when viewing a session hosted on another device.
• Use a different icon for "local on this device" vs "remote on another device".

Alternative Solutions

_No response_

Priority

Medium - Would be very helpful

Feature Category

Other

Use Case Example

_No response_

Additional Context

_No response_