Refresh: Update search context selector

Created by: umpox

Search Context selector

Before:

After (Redesign disabled):

After (Redesign enabled)

As most of these changes are purely layout, it makes sense to update these here and just trigger the color updates when the refresh is enabled.

Designs: https://www.figma.com/file/NIsN34NH7lPu04olBzddTw/Design-Refresh-Systemization-source-of-truth?node-id=954%3A2724

Issue: https://github.com/sourcegraph/sourcegraph/issues/20369