Support pluggable logging for structured output and correlation IDs

[maxisbey]

Summary

Allow users to inject their own logger for structured logging, correlation IDs, and integration with their logging infrastructure.

Problem

The SDK uses standard Python logging.getLogger(__name__) which produces unstructured, multi-line output that's hard to correlate in log aggregation systems.

• Can't correlate logs from a single request when debugging
• Multi-line log entries break standard log parsing
• No built-in support for correlation IDs across request lifecycle

Goal

Let users configure a custom logger factory so they can use structured logging (JSON), add correlation IDs, and integrate with tools like structlog.

---

_{AI Disclaimer}

[maxisbey]

Potentially related: #421

[maxisbey]

v2 went a different direction here, so this no longer applies as written — closing. Feel free to reopen if there's something still outstanding.