AI: Fix prompt lifecycle hooks not firing for `wp_ai_client_prompt()` flow.

This broke while porting over the relevant logic from the `wp-ai-client` package to Core.

With this changeset, the lifecycle hooks `wp_ai_client_before_generate_result` and `wp_ai_client_after_generate_result` fire as expected.

Props shadyvb, gziolo.
Fixes #64935.


git-svn-id: https://develop.svn.wordpress.org/trunk@62101 602fd350-edb4-49c9-b593-d223f7449a82