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/components/ProviderList.test.tsx

@@ -122,7 +122,7 @@ describe("ProviderList Component", () => {
       <ProviderList
         providers={{}}
         currentProviderId=""
-        appType="claude"
+        appId="claude"
         onSwitch={vi.fn()}
         onEdit={vi.fn()}
         onDelete={vi.fn()}
@@ -150,7 +150,7 @@ describe("ProviderList Component", () => {
       <ProviderList
         providers={{}}
         currentProviderId=""
-        appType="claude"
+        appId="claude"
         onSwitch={vi.fn()}
         onEdit={vi.fn()}
         onDelete={vi.fn()}
@@ -189,7 +189,7 @@ describe("ProviderList Component", () => {
       <ProviderList
         providers={{ a: providerA, b: providerB }}
         currentProviderId="b"
-        appType="claude"
+        appId="claude"
         isEditMode
         onSwitch={handleSwitch}
         onEdit={handleEdit}