docs: Gateway logo, docs switcher and llm-gateway links

[SebConejo]

Why

The docs now cover the Manifest LLM Gateway, and the self-healing product gets its own docs. Readers need to see which docs they are on and reach the other one.

What changed

• The sidebar shows the Manifest Gateway logo at the same 30px height.
• A click on the logo opens a menu with Manifest Gateway Docs (active) and Manifest Self-Healing Docs (https://manifest.build/docs).
• The theme switch sits at the bottom of the sidebar, under GitHub and Discord.
• The search dialog and its backdrop sit above the banner.
• The banner reads "NEW LAUNCH The self-healing layer for APIs" and the search prompt reads "Search in docs...".
• The page footer is gone. Its links and theme switch already live in the sidebar.
• Every repo link points to mnfst/llm-gateway, and the git clone examples use cd llm-gateway.
• The coherence check ignores the variables set on the user's machine for the CLI and MCP clients. The environment-variables reference lists server variables only.

Merged from #97

This branch includes #97, which moves the docs to /llm-gateway/docs. I kept its content and namespace, and this branch's banner, search prompt and repo links. The CLI and MCP pages merged on main after #97 was opened now also live under /llm-gateway/docs.

This PR replaces #100.

Check

node scripts/check-coherence.mjs is GREEN (98 pages).