[Testing] Implement logging verification for web runner

Migration from https://github.com/aws/aws-durable-execution-sdk-python-testing/issues/132

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## Overview

- We have implemented a customized logger with durable execution info (ref: https://github.com/aws/aws-durable-execution-sdk-python/blob/main/src/aws_durable_execution_sdk_python/logger.py#L55).
- We should be able to verify the logging works correctly with integration tests.

## AC

- Implement a handler that checks cloudwatch logs are created correctly
- Add logging verification for logger integration tests

从 src/aws_durable_execution_sdk_python/logger.py 中引用的自定义 logger 开始，并找到现有的 logger 集成测试。实现 CloudWatch 日志检查 handler，并在集成测试中添加 assertion，以确认 durable execution logging 已正确创建；当针对预期 CloudWatch 输出的 logging 验证通过时，即视为完成。

由索引模型根据 Issue 内容生成。