website: wire the site into the repo gates; make every tutorial example compile

- website joins the pnpm workspace; root scripts website:dev/website:build;
  run-gates gains a website-build gate (ci-primary + ci-static) — the
  VitePress build doubles as the site's dead-link check; AGENTS.md documents
  the commands.
- doc-typecheck + verify-type-equiv now scan website/zh-CN/**/*.md; every
  ```typescript fence converted to ```ts and made standalone-compilable
  (55 compiled, 1 ignore-check). Phantom APIs the compiler caught are fixed:
  invented event names (agent/turn-end, tool/call, llm/pre-request, ready,
  dispose) replaced with real catalog events or per-plugin declare-module
  merges; presentCall/inject/Config claims corrected to the real shapes.
- guide/config.md entry-fields table completed against loader EntryOptions;
  its coding-agent example brought in line with examples/coding-agent.

scripts/run-gates.ts

@@ -167,6 +167,7 @@ function ciPrimaryGates(): Gate[] {
     ...docSyncLeafGates(),
     pnpmScript('module-graph', 'verify-module-graph', { label: 'module graph' }),
     pnpmScript('knip', 'knip'),
+    pnpmScript('website-build', 'website:build', { label: 'website build' }),
     pnpmScript('build', 'build', { needs: ['typecheck'] }),
     pnpmScript('publint', 'publint', { needs: ['build'] }),
     pnpmScript('node-next-types', 'verify-node-next-types', {
@@ -184,6 +185,7 @@ function ciStaticGates(): Gate[] {
     ...docSyncLeafGates(),
     pnpmScript('module-graph', 'verify-module-graph', { label: 'module graph' }),
     pnpmScript('knip', 'knip'),
+    pnpmScript('website-build', 'website:build', { label: 'website build' }),
   ]
 }