[Integration Spec] Update integration-client.md

[chernser]

Summary

• Update integration-client.md
• Changes structure to be more consistent
• Reorganizes sections

Checklist

Delete items not relevant to your PR:

• Closes #
• Unit and integration tests covering the common scenarios were added
• A human-readable description of the changes was provided to include in CHANGELOG
• For significant changes, documentation in https://github.com/ClickHouse/clickhouse-docs was updated with further explanations or tutorials