feat(website): one-command CLI install + Segment install-intent events on MCP docs

[dylantientcheu]

What

Reworks the Install DocSearch MCP docs (docs/mcp/installation) so we can see who actually wants to install it, and gives the fastest path.

One-command install (design)

• New terminal-style One-command install section in MCPInstall, between the client picker and the step-by-step panel — npx @docsearch/cli setup --<client>, with the flag crossfading as you switch clients (mirrors the MCP frontend hero).
• Renders clean for clients the CLI actually supports (cursor, claude, codex, opencode — verified against packages/docsearch-cli flags) and shows default for the rest.
• installation.mdx now introduces the CLI as the fastest path, with plugin/manual as the do-it-yourself routes.

Segment install-intent events

• Adds @segment/analytics-next behind a small lazy-loaded, SSR-safe helper (src/lib/segment.js) — Segment only loads on the first tracked interaction.
• Events (same source as the MCP frontend, so both sites funnel together; context.page tells them apart):
  • Install Command Copied — CLI one-liner copy (client, command)
  • Install Config Copied — any config code-card copy (client, mode, caption)
  • Install Button Clicked — Cursor one-click deeplink (client, mode)

Why based on feat/docusaurus-adapter-v5

The website only builds with the migrated themeConfig shape from #2904, so this stacks on top of it and should merge after it.

Test plan

• bun run build in packages/website — client + SSR compile, static build succeeds
• Not yet manually exercised in a browser — worth a quick pass over the picker (flag follows the selected client, section collapses for unsupported clients) and the Segment debugger before merging

🤖 Generated with Claude Code

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[vascobettencourt]

The quick-install path needs to be limited to the clients the CLI can actually configure. CI is green, but the current UI advertises a nonfunctional install command for every unsupported client.

[vascobettencourt]

This AnimatePresence always receives a child, so the quick-install panel never exits for unsupported clients. Selecting ChatGPT, Pi, VS Code, Windsurf, Zed, Conductor, Antigravity, or Other still shows npx @docsearch/cli setup and claims it adds the server/rules/skills to that selected client. The unflagged CLI only detects/prompts for Cursor, Claude, Codex, and OpenCode, so it cannot configure the selected unsupported client. Please gate the child on CLI_SUPPORTED_CLIENTS.has(selected.id) (which will also let this exit animation run), as described in the PR summary.

[dylantientcheu]

by default the setup command installs for most agents in the relevant folders.

those that are not listed is because it will justwork™️ without any specific install instructions