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/storage/stores/settings-store.ts

@@ -0,0 +1,34 @@
+import { Store } from "../store.js";
+import type { StoreConfig } from "../types.js";
+
+/**
+ * Store for application settings (theme, proxy config, etc.).
+ */
+export class SettingsStore extends Store {
+	getConfig(): StoreConfig {
+		return {
+			name: "settings",
+			// No keyPath - uses out-of-line keys
+		};
+	}
+
+	async get<T>(key: string): Promise<T | null> {
+		return this.getBackend().get("settings", key);
+	}
+
+	async set<T>(key: string, value: T): Promise<void> {
+		await this.getBackend().set("settings", key, value);
+	}
+
+	async delete(key: string): Promise<void> {
+		await this.getBackend().delete("settings", key);
+	}
+
+	async list(): Promise<string[]> {
+		return this.getBackend().keys("settings");
+	}
+
+	async clear(): Promise<void> {
+		await this.getBackend().clear("settings");
+	}
+}