`ErrorBadgeManager` rewrite to separate inspector specific logic

[elliette]

ErrorBadgeManager has a lot of conditional logic for inspector errors (e.g., appendError only works for inspector errors because we are only saving inspector errors in the `activeErrors map:

devtools/packages/devtools_app/lib/src/shared/managers/error_badge_manager.dart

Lines 32 to 37 in 36ac0a2

	final _activeErrors =
	<String, ValueNotifier<LinkedHashMap<String, DevToolsError>>>{
	InspectorScreen.id: ValueNotifier<LinkedHashMap<String, DevToolsError>>(
	LinkedHashMap<String, DevToolsError>(),
	),
	};

)

This file and logic could use an overhaul to move the Inspector specific logic to the Inspector controller.