[BUG] Add Claude Code icon to Activity bar

Preflight Checklist

• [x] I have searched existing issues and this hasn't been reported yet
• [x] This is a single bug report (please file separate reports for different bugs)
• [x] I am using the latest version of Claude Code

What's Wrong?

The problem described by #17124 is not solved and the task is closed.
Please reopen and implement the function.

What Should Happen?

Claude code icon should appear on the Activity Bar to quickly launch a separate window with claude code terminal or chat depending on internal settings

Error Messages/Logs

Steps to Reproduce

<img width="435" height="694" alt="Image" src="https://github.com/user-attachments/assets/6ffed330-fe53-467d-ba15-d72e78aefeaf" />

<img width="1698" height="831" alt="Image" src="https://github.com/user-attachments/assets/dd6283e1-4184-4d46-b017-41ef1e99729b" />

<img width="683" height="322" alt="Image" src="https://github.com/user-attachments/assets/65fb7460-402a-4a45-948e-a5667ed07569" />

Claude Model

None

Is this a regression?

Yes, this worked in a previous version

Last Working Version

_No response_

Claude Code Version

v2.1.23

Platform

Anthropic API

Operating System

macOS

Terminal/Shell

VS Code integrated terminal

Additional Information

_No response_