docs: unify badge style

[CosmDandy]

Every badge now goes through shields.io with one style, so the row reads as a single system instead of three services with different heights and corner radii.

• Uniform grey label on the left (labelColor=21262d) across every badge.
• Colour carries meaning on the right: live status keeps semantic green/red, static badges use the purple and cyan from the og image palette.
• Links are reference-style, keeping long URLs out of the body.
• The Codespaces button stays above the row, where it reads as the call to action.