Fix persistent and disappearing error badges for Inspector

[khanak0509]

issue #9524
This PR fixes issue #9524 where the Inspector error badge was acting weird — like it kept increasing incorrectly and sometimes disappeared on tab switch. The count was blindly incremented, errors were getting deleted due to aggressive filtering, and the badge was cleared too early.

How I fixed it:
Basically, I made the badge count derived from actual unread errors instead of manually incrementing it, removed the aggressive filtering, and stopped clearing the badge on tab switch — like now it only resets on a fresh connection or Hot Restart.
before

before.mov

after

after.mov

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[gemini-code-assist]

Code Review

This pull request refactors the ErrorBadgeManager to use a derived-count model for the Inspector, where the badge count is automatically updated based on the number of unread errors. This change removes the need for manual badge count increments and the filterErrors logic in the InspectorController. A concern was raised regarding redundant calls to clearErrorCount remaining in the InspectorController files, which should be removed to maintain code clarity and align with the new error tracking model.