Refactor API: Remove 'any' from EmbeddedChatApi and improve payload typing

Summary

packages/api/src/EmbeddedChatApi.ts currently relies on multiple explicit any types across callback signatures, stream payload handling, auth inputs, and message APIs. This reduces type safety and makes it easier to accidentally access unsafe or undefined fields, especially when working with dynamic DDP payloads.

Problem

• Explicit any appeared in:
  • message listeners
  • action/ui interaction listeners
  • autoLogin credentials
  • DDP stream callbacks (onStreamData)
  • sendMessage input
  • handleUiKitInteraction input
• Stream payloads were accessed without defensive narrowing (fields, eventName, args).
• SDK callback types are broad, so local guards are needed to keep runtime behavior while improving static safety.

Acceptance Criteria

• No explicit any remains in packages/api/src/EmbeddedChatApi.ts.
• connect() stream handling uses guarded/narrowed access.
• sendMessage and handleUiKitInteraction use typed inputs.
• packages/api builds successfully.
• No lint/type errors introduced.

Impact

• Better type safety and editor intelligence.
• Lower risk of runtime errors from malformed DDP payloads.
• No intended behavior changes at runtime.

Start with packages/api/src/EmbeddedChatApi.ts, reading connect(), the callback signatures, autoLogin, sendMessage, and handleUiKitInteraction. Run the packages/api build and lint/type checks while replacing explicit any with typed inputs and guarded stream access; done means no explicit any remains and the package builds without new errors.