Feature request: persistent indicator when in Claude Code vs Claude.ai chat

### 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

As a non-technical user, I ended up in Claude Code (via the desktop app's Code tab)
without realizing I'd left the regular Claude.ai chat experience. There's currently
nothing prominent distinguishing the two, so it wasn't obvious I'd switched contexts —
I only figured it out after asking about missing chat history.

### Proposed Solution

Suggestion: a persistent, visible banner or label at the top of the screen when in
Claude Code, e.g. "You are in Claude Code," so it's unmistakable which product/mode
a user is in.

### Alternative Solutions

Different coloured tab / window

### Priority

Medium - Would be very helpful

### Feature Category

Other

### Use Case Example

_No response_

### Additional Context

_No response_

No files, tests, or entry points are named. Start by locating the desktop app's Code-tab UI entry point and clarifying whether the indicator belongs to Claude Code or the host app. Done means users can continuously and unmistakably tell when they are in Claude Code rather than Claude.ai chat.

Rédigé par le modèle d'indexation à partir du texte de l'issue.