chore(apigateway): enhance code arounds method authorizers (#30978)

### Issue # (if applicable)

### Reason for this change

Add comments and enhance the code around method authorizers.

### Description of changes

No behavioural change, should only include comments, variables name update and re-ordering.

### Description of how you validated changes

All existing tests pass.

### Checklist
- [ ] My code adheres to the [CONTRIBUTING GUIDE](https://github.com/aws/aws-cdk/blob/main/CONTRIBUTING.md) and [DESIGN GUIDELINES](https://github.com/aws/aws-cdk/blob/main/docs/DESIGN_GUIDELINES.md)

----

*By submitting this pull request, I confirm that my contribution is made under the terms of the Apache-2.0 license*

packages/aws-cdk-lib/aws-apigateway/lib/method.ts

@@ -185,22 +185,35 @@ export class Method extends Resource {
     validateHttpMethod(this.httpMethod);
 
     const options = props.options || {};
-
     const defaultMethodOptions = props.resource.defaultMethodOptions || {};
+
     // do not use the default authorizer config in case if the provided authorizer type is None
     const authorizer =
         options.authorizationType === AuthorizationType.NONE
         && options.authorizer == undefined ? undefined : options.authorizer || defaultMethodOptions.authorizer;
     const authorizerId = authorizer?.authorizerId ? authorizer.authorizerId : undefined;
 
-    const authorizationTypeOption = options.authorizationType || defaultMethodOptions.authorizationType;
-    const authorizationType = authorizer?.authorizationType || authorizationTypeOption || AuthorizationType.NONE;
-
-    // if the authorizer defines an authorization type and we also have an explicit option set, check that they are the same
-    if (authorizer?.authorizationType && authorizationTypeOption && authorizer?.authorizationType !== authorizationTypeOption) {
-      throw new Error(`${this.resource}/${this.httpMethod} - Authorization type is set to ${authorizationTypeOption} ` +
-        `which is different from what is required by the authorizer [${authorizer.authorizationType}]`);
-    }
+    /**
+     * Get and validate authorization type from the values set by API resource and method.
+     *
+     * REST API Resource
+     * └── defaultMethodOptions: Method options to use as a default for all methods created within this API unless custom options are specified.
+     *    ├── authorizationType: Specifies the default authorization type unless custom options are specified, recommended to not be specified.
+     *    └── authorizer: Specifies the default authorizer for all methods created within this API unless custom options are specified.
+     *        └── authorizerType: The default authorization type of this authorizer.
+     *
+     * REST API Method
+     * └── options: Method options.
+     *    ├── authorizationType: Specifies the authorization type, recommended to not be specified.
+     *    └── authorizer: Specifies an authorizer to use for this method.
+     *        └── authorizerType: The authorization type of this authorizer.
+     *
+     * Authorization type is first set to "authorizer.authorizerType", falling back to method's "authorizationType",
+     * falling back to API resource's default "authorizationType", and lastly "Authorizer.NONE".
+     *
+     * Note that "authorizer.authorizerType" should match method or resource's "authorizationType" if exists.
+     */
+    const authorizationType = this.getMethodAuthorizationType(options, defaultMethodOptions, authorizer);
 
     // AuthorizationScope should only be applied to COGNITO_USER_POOLS AuthorizationType.
     const defaultScopes = options.authorizationScopes ?? defaultMethodOptions.authorizationScopes;
@@ -303,6 +316,27 @@ export class Method extends Resource {
     this.methodResponses.push(methodResponse);
   }
 
+  /**
+   * Get API Gateway Method's authorization type
+   * @param options API Gateway Method's options to use
+   * @param defaultMethodOptions API Gateway resource's default Method's options to use
+   * @param authorizer Authorizer used for API Gateway Method
+   * @returns API Gateway Method's authorizer type
+   */
+  private getMethodAuthorizationType(options: MethodOptions, defaultMethodOptions: MethodOptions, authorizer?: IAuthorizer): string {
+    const authorizerAuthType = authorizer?.authorizationType;
+    const optionsAuthType = options.authorizationType || defaultMethodOptions.authorizationType;
+    const finalAuthType = authorizerAuthType || optionsAuthType || AuthorizationType.NONE;
+
+    // if the authorizer defines an authorization type and we also have an explicit option set, check that they are the same
+    if (authorizerAuthType && optionsAuthType && authorizerAuthType !== optionsAuthType) {
+      throw new Error(`${this.resource}/${this.httpMethod} - Authorization type is set to ${optionsAuthType} ` +
+        `which is different from what is required by the authorizer [${authorizerAuthType}]`);
+    }
+
+    return finalAuthType;
+  }
+
   private renderIntegration(bindResult: IntegrationConfig): CfnMethod.IntegrationProperty {
     const options = bindResult.options ?? {};
     let credentials;