Refactor ManagedAPIServer in examples into a shared utility

[xingyaoww]

Summary

The ManagedAPIServer context manager class is duplicated across multiple example files in examples/02_remote_agent_server/:

• 01_convo_with_local_agent_server.py
• 11_conversation_fork.py
• 12_settings_and_secrets_api.py
• 13_workspace_get_llm.py

Each file contains its own copy of the class, which is a context manager that launches a subprocess-managed OpenHands API server with a randomly generated session API key for secure access.

Proposed Change

Extract ManagedAPIServer into a shared utility module (e.g., examples/02_remote_agent_server/_utils.py or similar) and import it from each example. This would:

1. Reduce duplication — single source of truth for the managed server logic
2. Make maintenance easier — bug fixes or enhancements apply everywhere at once
3. Keep examples focused — example scripts can focus on demonstrating the SDK feature rather than boilerplate server management

Notes

• The implementations vary slightly across examples (e.g., different health check logic, environment setup), so the shared version should be flexible enough to accommodate those differences.
• Care should be taken to keep examples self-explanatory — the shared util should be well-documented so readers can still understand what's happening.

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This issue was created by an AI agent (OpenHands) on behalf of @xingyaoww, based on this PR comment.