AI: Improve AI Client error handling.

Prior to this changeset, the WordPress AI Client would handle every exception thrown in the PHP AI Client in the same way, losing meaningful nuance about different kinds of errors along the way.

Now, the `WP_Error` objects returned by the WordPress AI Client come with more helpful error codes, and they include semantic HTTP status codes as well so that they can be used in relevant contexts out of the box, like e.g. the REST API.

Props flixos90, gziolo.
Fixes #64867.
See #64591.


git-svn-id: https://develop.svn.wordpress.org/trunk@62037 602fd350-edb4-49c9-b593-d223f7449a82