fix(auth): handle custom Cognito domains without appending regional suffix

.changeset/eighty-feet-go.md

@@ -0,0 +1,5 @@
+---
+'@aws-amplify/backend-auth': patch
+---
+
+Add unit tests for custom domain and cognito-managed domain OAuth scenarios

.changeset/honest-chairs-rescue.md

@@ -0,0 +1,2 @@
+---
+---

packages/backend-auth/src/lambda/reference_auth_initializer.test.ts

@@ -311,6 +311,38 @@ void describe('ReferenceAuthInitializer', () => {
     );
   });
 
+  void it('handles custom domain with external login providers', async () => {
+    describeUserPoolResponse = {
+      ...httpSuccess,
+      UserPool: {
+        ...UserPool,
+        CustomDomain: 'auth.dev.example.com',
+      },
+    };
+    const result = await handler.handleEvent(createCfnEvent);
+    assert.strictEqual(result.Status, 'SUCCESS');
+    assert.ok(result.Data);
+    assert.strictEqual(result.Data.oauthCognitoDomain, 'auth.dev.example.com');
+  });
+
+  void it('handles cognito-managed domain with external login providers', async () => {
+    describeUserPoolResponse = {
+      ...httpSuccess,
+      UserPool: {
+        ...UserPool,
+        CustomDomain: undefined,
+        Domain: 'ref-auth-userpool-1',
+      },
+    };
+    const result = await handler.handleEvent(createCfnEvent);
+    assert.strictEqual(result.Status, 'SUCCESS');
+    assert.ok(result.Data);
+    assert.strictEqual(
+      result.Data.oauthCognitoDomain,
+      'ref-auth-userpool-1.auth.us-east-1.amazoncognito.com',
+    );
+  });
+
   void it('throws if user pool group is not found', async () => {
     listGroupsResponse = {
       ...httpSuccess,

packages/backend-auth/src/lambda/reference_auth_initializer.ts

@@ -459,7 +459,9 @@ export class ReferenceAuthInitializer {
 
     // domain
     const oauthDomain = userPool.CustomDomain ?? userPool.Domain ?? '';
-    const fullDomainPath = `${oauthDomain}.auth.${region}.amazoncognito.com`;
+    const fullDomainPath = userPool.CustomDomain
+      ? userPool.CustomDomain
+      : `${oauthDomain}.auth.${region}.amazoncognito.com`;
     const data = {
       signupAttributes: JSON.stringify(
         userPool.SchemaAttributes?.filter(