Fix reference docs

[pquentin]

• Add Async Search that got removed by mistake in a previous refactor
• Fix warnings caused by a typo in no-index
• Prefer elasticsearch.client over elasticsearch for consistency
• Last remaining warning will be fixed by Fix syntax in ClusterHealthRequest elasticsearch-specification#2768

[github-actions]

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[szabosteve]

Thank you for these changes! 🙇 LGTM!