[FEATURE] Desktop app: show a local/remote (SSH host) indicator on projects and sessions in the sidebar

### Preflight Checklist

- [x] I have searched [existing requests](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20label%3Aenhancement) and this feature hasn't been requested yet
- [x] This is a single feature request (not multiple features)

### Problem Statement

In the Claude Desktop app (Code tab), the sidebar groups sessions by project folder **name only**. A local folder `D:\work\reports` and an SSH-remote folder `/data/share/reports` both appear as a group called `reports`, with identical icons and identical session rows. There is no way to tell at a glance which projects/sessions run on my local machine and which run on a remote host over SSH.

This is not cosmetic. The machines do not share files or environments: a path, conda env, or dataset that exists on the server does not exist locally (and vice versa). Picking the wrong `reports` group means Claude edits files on the wrong machine or runs commands in the wrong environment. I have to open a session and inspect the working directory or run `pwd` before I can trust it.

Two earlier requests for exactly this were auto-closed as inactive / not planned, not because the need went away:
- #67086 "Show remote SSH host indicator in Claude Desktop app"
- #83170 "Make it visible whether a session runs locally or remotely (session list indicator)"

Reopening the request with a concrete UI proposal.

### Proposed Solution

Minimal version (biggest win for least work):
1. In the sidebar, render a small badge/icon next to each **project group header** and each **session row** indicating where it runs: e.g. a monitor icon or `Local` for local folders, a cloud/server icon plus the SSH host alias (`user@gpu-node`, or the `Host` name from `~/.ssh/config`) for remote ones.
2. On hover over a group header, show the full absolute path and host in a tooltip.

Nice-to-have:
3. Allow grouping/sorting the sidebar by host (Local / host-A / host-B) instead of only by folder name, or at least colour-code groups per host.
4. Include the host in the auto-generated session title prefix, or expose a per-project display-name override so users can label groups themselves.

The desktop app already knows which sessions are SSH remote (it manages the SSH connection and auto-installs Claude Code on the host), so the information exists; it just is not surfaced.

### Alternative Solutions

- Renaming sessions by hand with `/rename [srv] ...` — works per session only, does not fix the group header, and is forgotten easily.
- Creating prefixed symlinks on the server (`srv-reports -> reports`) so the folder name itself carries the marker — hacky, pollutes the server filesystem, breaks for shared team directories.
- Putting a CLAUDE.md in each directory that says "this directory is on server X" — helps Claude, does not help the human reading the sidebar.
- Hooks/plugins cannot help: plugins are limited to commands/agents/MCP/hooks with no sidebar UI extension point, and the statusline is terminal-only.
- Other tools: OpenAI Codex's desktop app shows a clear local vs remote indicator on each project/session; VS Code Remote-SSH shows the host in the status bar and window title. Users coming from those tools expect the same.

### Priority

High - Significant impact on productivity

### Feature Category

Interactive mode (TUI)

### Use Case Example

1. I am a computational biologist. I keep analysis projects both on my Windows laptop and on a Linux GPU server reached via SSH.
2. Several projects have the same folder name on both machines (`reports`, `analysis`, `data`), because the server copy is the production run and the local copy is a scratch/preview.
3. In the desktop sidebar I see two groups called `reports`. I click one, ask Claude to regenerate a figure, and it runs on the laptop against a stale copy — or worse, on the server against production data when I only wanted a preview.
4. With a host badge on the group header I would pick the right one instantly and would not have to run `pwd`/`hostname` at the start of every session.

### Additional Context

- Related closed issues: #67086, #83170 (both closed as not-planned/inactive; problem persists).
- Related: #78595 asks for other session-list status indicators; a host indicator fits naturally into the same UI slot.
- Platform: Claude Desktop on Windows 11, Code tab, mix of local projects and SSH remote sessions.
- I am happy to test a build or provide screenshots of the current ambiguous sidebar.

No implementation files or tests are named. Start by locating the Claude Desktop Code-tab sidebar and the SSH session metadata used by the app, then trace how project groups and session rows are rendered. Done means local and remote projects and sessions visibly identify their host, with the full path and host available on hover.

Written by the indexing model from the issue text.