Refactor to Store-based architecture

- Create base Store class with private backend and protected getBackend()
- Add SettingsStore, ProviderKeysStore, SessionsStore
- Each store defines its own schema via getConfig()
- AppStorage now takes stores + backend in constructor
- Remove SessionsRepository (logic moved to SessionsStore)
- Update all consumers to use store API (storage.settings.get/set, storage.providerKeys.get/set)
- Update example app to follow new pattern: create stores, gather configs, create backend, wire
- Benefits: stores own their schema, no circular deps, cleaner separation

packages/web-ui/src/dialogs/ApiKeyPromptDialog.ts

@@ -29,7 +29,7 @@ export class ApiKeyPromptDialog extends DialogBase {
 
 		// Poll for key existence - when key is added, resolve and close
 		const checkInterval = setInterval(async () => {
-			const hasKey = !!(await getAppStorage().getProviderKey(this.provider));
+			const hasKey = !!(await getAppStorage().providerKeys.get(this.provider));
 			if (hasKey) {
 				clearInterval(checkInterval);
 				if (this.resolvePromise) {