Apply headful plugin host logging strategy to headless host

[colin-grant-work]

What it does

Applies the plugin-side log-level tracking introduced in #14908 to the headless host, so that, e.g. console.debug in an extension is treated as a debug-level log in the application logs.

How to test

1. Run the application with headless extension that uses console logging at various levels.
2. Confirm that the logs show up in the application logs with the correct level prefix, and only if the specified level is enabled.

Follow-ups

Breaking changes

• This PR introduces breaking changes and requires careful review. If yes, the breaking changes section in the changelog has been updated.

Attribution

Review checklist

• As an author, I have thoroughly tested my changes and carefully followed the review guidelines
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Reminder for reviewers

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