Improve AI Client error handling

[felixarntz]

See the Trac ticket description for what this PR does.

Trac ticket: https://core.trac.wordpress.org/ticket/64867

Use of AI Tools

Claude Opus was used to help with tests.

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This Pull Request is for code review only. Please keep all other discussion in the Trac ticket. Do not merge this Pull Request. See GitHub Pull Requests for Code Review in the Core Handbook for more details.

[github-actions]

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Props flixos90.

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[felixarntz]

Note for reviewers: This class doesn't exist, it was pointless and confusing to include this reference here just for consistency. Unlike the other WP_Error objects, this one is not based on an exception.

[github-actions]

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