refactor(types): rename AppType to AppId for semantic clarity

Rename `AppType` to `AppId` across the entire frontend codebase to better
reflect its purpose as an application identifier rather than a type category.
This aligns frontend naming with backend command parameter conventions.

Changes:
- Rename type `AppType` to `AppId` in src/lib/api/types.ts
- Remove `AppType` export from src/lib/api/index.ts
- Update all component props from `appType` to `appId` (43 files)
- Update all variable names from `appType` to `appId`
- Synchronize documentation (CHANGELOG, refactoring plans)
- Update test files and MSW mocks

BREAKING CHANGE: `AppType` type is no longer exported. Use `AppId` instead.
All component props have been renamed from `appType` to `appId`.

tests/integration/App.test.tsx

@@ -41,13 +41,13 @@ vi.mock("@/components/providers/ProviderList", () => ({
 }));
 
 vi.mock("@/components/providers/AddProviderDialog", () => ({
-  AddProviderDialog: ({ open, onOpenChange, onSubmit, appType }: any) =>
+  AddProviderDialog: ({ open, onOpenChange, onSubmit, appId }: any) =>
     open ? (
       <div data-testid="add-provider-dialog">
         <button
           onClick={() =>
             onSubmit({
-              name: `New ${appType} Provider`,
+              name: `New ${appId} Provider`,
               settingsConfig: {},
               category: "custom",
               sortIndex: 99,