[FEATURE] Add visual distinction between Chat Projects and Cowork Projects in sidebar

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

Add visual distinction between Chat Projects and Cowork Projects in sidebar
Description:
The sidebar lists Chat Projects and Cowork Projects under the same "Projects" section with identical visual appearance. There is no label, icon, badge, or section separator to distinguish them.
Current behavior:

Both project types appear identically in the list. The only way to determine the type is to open the project.
Problem this causes:

When a Chat Project and a Cowork Project share the same name (which happens naturally when users mirror their Chat project structure in Cowork), it is impossible to tell them apart from the list view.
Requested behavior:

A small visual indicator on each project entry showing its type — for example a label ("Chat" / "Cowork"), a distinct icon, or a section separator in the sidebar.
App version: Claude for Mac 1.15962.1

Proposed Solution

A small visual indicator on each project entry showing its type — for example a label ("Chat" / "Cowork"), a distinct icon, or a section separator in the sidebar.

Alternative Solutions

Go Back to 3 tabs CHAT/COWORK/CODE

Priority

High - Significant impact on productivity

Feature Category

Configuration and settings

Use Case Example

_No response_

Additional Context

_No response_