Web IDE context header redesign

What does this MR do?

Implements the redesign for the Web IDE context header and back button.

NOTE:

• This redesign is coupled to the new merge request / branch picker. This will be handled in a separate MR pointing to this branch.
• This MR will be WIP until both the back button and branch picker are implemented.

Are there points in the code the reviewer needs to double check?

Why was this MR needed?

• #47768 (closed)
• #46165 (closed)

Screenshots (if relevant)

Does this MR meet the acceptance criteria?

• Changelog entry added, if necessary
• [ ] Documentation created/updated
• [ ] API support added
• Tests added for this feature/bug
• Conforms to the code review guidelines
  • Has been reviewed by a UX Designer
  • Has been reviewed by a Frontend maintainer
  • [ ] Has been reviewed by a Backend maintainer
  • [ ] Has been reviewed by a Database specialist
• Conforms to the merge request performance guidelines
• Conforms to the style guides
• Conforms to the database guides
• If you have multiple commits, please combine them into a few logically organized commits by squashing them
• Internationalization required/considered
• End-to-end tests pass (package-and-qa manual pipeline job)

What are the relevant issue numbers?

Closes #47768 (closed)