Persist logged-out chat sessions with a secure browser principal

## Problem
Frontend Agent Chat can now chat with public/audience-granted agents while logged out, but session history does not persist safely for anonymous visitors. The UI enables sessions, but the server-side session APIs currently require a principal that maps to a WordPress user.

We should not use the insecure middle ground of letting `audience:public` list sessions owned by the agent owner or by the whole public audience.

## Required shape
- Bootstrap an opaque browser principal for logged-out visitors.
- Prefer an HttpOnly, Secure, SameSite cookie so the secret is not exposed to JavaScript.
- Send session list/get/delete requests under that browser principal.
- Keep active agent preference working for logged-out visitors without depending on user options.
- Add a visible path for future account persistence, e.g. “Sign in with WordPress.com to save chat history.”

## Acceptance criteria
- Logged-out visitors can reload the page and see their own previous chat sessions.
- A different browser/session cannot list or read those sessions.
- Logged-in users continue to use normal user-owned session persistence.
- No raw `session_id` is treated as the sole authorization secret.
- The UI behaves gracefully when browser cookies are disabled.

## Related work
- Agents API needs principal-owned conversation sessions.
- Data Machine needs transcript ownership separate from runtime execution user.
- A8C Intelligence can implement WordPress.com login/account-linking for persistence across devices.

Start by tracing the server-side session APIs and the logged-out UI request path, then inspect how principals, WordPress users, and active-agent preferences are currently handled. Done means anonymous sessions survive reloads but remain isolated by browser principal, authenticated persistence is unchanged, raw session IDs do not authorize access, and cookie-disabled browsers degrade gracefully.

Written by the indexing model from the issue text.