docs(kit): add client script & client context page

[antfubot]

Description

Adds a new DevTools Kit doc page — Client Script & Client Context (/kit/client-context) — introducing the two client-side foundations of embedded mode:

• The client script (@vitejs/devtools/client/inject): what it does when it runs in the host page (RPC connect, build context, publish global, mount the dock), how injection works via transformIndexHtml and virtual:vite-devtools-injection, and when it applies (dev serve, non-SSR, top-level windows only).
• The client context (DevToolsClientContext): a property overview (rpc, clientType, docks, panel, commands, when), access via getDevToolsClientContext(), the DockClientScriptContext extras for dock scripts, and how iframe panels connect instead.
• Troubleshooting: what to check when the client script isn't injected (backend integration, middleware mode, JS-only entries), with the manual import '@vitejs/devtools/client/inject' escape hatch, the double-mount caveat, and a dev-only import guard.

Also registers the page in the VitePress sidebar and the DevTools Kit nav dropdown, after "Dock System".

Verified with a full pnpm build + vitepress build.

Linked Issues

Additional context

Created with the help of an agent.

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commit: 6ed97e4