feat(api-gateway): deploy RestApi to existing stage (#29486)

### Issue #

Closes #25582 .

### Reason for this change

If I have to manually create a new deployment for API gateway but want to deploy to an existing stage instead of creating a new stage, the current workaround is `(deployment as any).resource.stageName = "myStage";`

### Description of changes

Added a new property `stageName` to `Deployment`. Updated README with two new sections: `Deploying to an existing stage` and `Controlled triggering of deployments` to show how users can trigger new deployments with their latest API changes if they manually create a `Deployment` resource for RestApi. 

Since `RestApiBase` manages `Deployment` constructs, this change applies to both `SpecRestApi` and `RestApi`. 

### Description of how you validated changes

Added unit tests and integration tests. Also successfully deployed a sample app to my account where I created a stage in the API gateway console and specified the stage name in the new deployment.

### Checklist
- [x] 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-testing/framework-integ/test/aws-apigateway/test/integ.spec-restapi.import-deploymentstage.js.snapshot/integtest-specrestapi-import-deployment-stage.template.json

@@ -0,0 +1,100 @@
+{
+ "Resources": {
+  "myapi4C7BF186": {
+   "Type": "AWS::ApiGateway::RestApi",
+   "Properties": {
+    "Body": {
+     "openapi": "3.0.2",
+     "info": {
+      "version": "1.0.0",
+      "title": "Test API for CDK"
+     },
+     "paths": {
+      "/pets": {
+       "get": {
+        "summary": "Test Method",
+        "operationId": "testMethod",
+        "responses": {
+         "200": {
+          "description": "A paged array of pets",
+          "content": {
+           "application/json": {
+            "schema": {
+             "$ref": "#/components/schemas/Empty"
+            }
+           }
+          }
+         }
+        },
+        "x-amazon-apigateway-integration": {
+         "responses": {
+          "default": {
+           "statusCode": "200"
+          }
+         },
+         "requestTemplates": {
+          "application/json": "{\"statusCode\": 200}"
+         },
+         "passthroughBehavior": "when_no_match",
+         "type": "mock"
+        }
+       }
+      }
+     },
+     "components": {
+      "schemas": {
+       "Empty": {
+        "title": "Empty Schema",
+        "type": "object"
+       }
+      }
+     }
+    },
+    "Name": "my-api"
+   }
+  },
+  "MyManualDeployment92F2175C1dcdcb8f1c24d86b6090e78df1fafcd3": {
+   "Type": "AWS::ApiGateway::Deployment",
+   "Properties": {
+    "RestApiId": {
+     "Ref": "myapi4C7BF186"
+    },
+    "StageName": "myStage"
+   }
+  }
+ },
+ "Parameters": {
+  "BootstrapVersion": {
+   "Type": "AWS::SSM::Parameter::Value<String>",
+   "Default": "/cdk-bootstrap/hnb659fds/version",
+   "Description": "Version of the CDK Bootstrap resources in this environment, automatically retrieved from SSM Parameter Store. [cdk:skip]"
+  }
+ },
+ "Rules": {
+  "CheckBootstrapVersion": {
+   "Assertions": [
+    {
+     "Assert": {
+      "Fn::Not": [
+       {
+        "Fn::Contains": [
+         [
+          "1",
+          "2",
+          "3",
+          "4",
+          "5"
+         ],
+         {
+          "Ref": "BootstrapVersion"
+         }
+        ]
+       }
+      ]
+     },
+     "AssertDescription": "CDK bootstrap stack version 6 required. Please run 'cdk bootstrap' with a recent version of the CDK CLI."
+    }
+   ]
+  }
+ }
+}