Wire the sidebar listChanged indicator across all screens #1402

Summary

The listChanged sidebar indicator badge is hardcoded false for every screen — Tools, Apps, Prompts, and Resources — in clients/web/src/components/views/InspectorView/InspectorView.tsx. It is never wired to a real notifications/*/list_changed signal, so the indicator can never light up.

Note this is the indicator badge only: the lists themselves already refresh live. For example, the managed-tools layer (core/react/useManagedTools.ts) updates the tools array on its toolsChange event, so the Apps/Tools sidebar contents stay current. What's missing is the visual "the server changed its list since you last looked" affordance.

This was split out of #1370 (render MCP Apps end-to-end), which deliberately kept scope to the apps render path.

Proposed work

• Surface a per-primitive "list changed since last refresh" boolean from the managed-state layer (tools/prompts/resources) up through App.tsx → InspectorView → each screen's listChanged prop.
• Decide the reset semantics (e.g. clear the flag when the user refreshes that list, or when the list is viewed).
• Apply consistently to all four screens so behavior is uniform.

Acceptance criteria

• A notifications/tools/list_changed (and prompts/resources equivalents) sets the corresponding screen's listChanged indicator.
• The indicator clears on the defined reset action.
• All four screens (Tools, Apps, Prompts, Resources) behave consistently; no screen is left hardcoded false.
• Tests cover the set + clear transitions.