Fix logging example in docs to include required StreamHandler

[jnowakowski]

The logging example in the documentation previously set the log level but did not include a handler, which can result in no output being shown in many environments (e.g., terminal, script execution).

This PR improves the example by:

• Adding a StreamHandler configured to write to sys.stdout
• Including a formatter to make the output more readable

This change improves developer experience, especially for those new to the library or debugging API calls.

• Make sure to open an issue as a bug/issue before writing your code! That way we can discuss the change, evaluate designs, and agree on the general idea
• Ensure the tests and linter pass
• Code coverage does not decrease (if any source code was changed)
• Appropriate docs were updated (if necessary)

Fixes #2604

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