integ-runner: --watch flag creates a cdk.context.json with `watch` field

### Describe the bug

integ runner creates a `cdk.context.json` file with the `watch` field if the the context file does not exist, when it actually needs a `cdk.json` with the `watch` field.

### Expected Behavior

Create a `cdk.json` with the `watch` field if it doesn't exist

### Current Behavior

integ runner creates a `cdk.context.json` file with the `watch` field if the the context file does not exist

### Reproduction Steps

1. Create a new application

````
mkdir bug
cdk init app --language=typescript
````

2. Add `integ-runner` and `integ-tests-alpha` to the `package.json`
``` json
"dependencies": {
"aws-cdk-lib": "2.144.0",
"constructs": "^10.0.0",
"@aws-cdk/integ-runner": "2.144.0-alpha.0",
"@aws-cdk/integ-tests-alpha": "2.144.0-alpha.0",
"source-map-support": "^0.5.21"
}
}
```

3. `npm install`
4. create the following `test/integ.example.test.ts` file
```ts
import * as integ from '@aws-cdk/integ-tests-alpha';
import * as cdk from 'aws-cdk-lib';
import { Construct } from "constructs";

const app = new cdk.App();

class TestStack extends cdk.Stack {
constructor(scope: Construct, id: string) {
super(scope, id);
new cdk.CfnWaitConditionHandle(this, 'NullResource');
}
}
const stack = new TestStack(app, 'TestStack');
new integ.IntegTest(app, 'integration-test', {
testCases: [stack],
});
```
5. Optional: delete the `cdk.json` file (doesn't matter)
5. npx integ-runner test/integ.example.test.ts --watch

### Possible Solution

1. Don't create a `cdk.context.json`
2. create a /update the existing `cdk.json` with the `watch` field

### Additional Information/Context

The documentation needs to be update to reflect this

### CDK CLI Version

2.144.0 (build 5fb15bc)

### Framework Version

_No response_

### Node.js Version

v18.15.0

### OS

Osx

### Language

TypeScript

### Language Version

_No response_

### Other information

_No response_

Start with the integ-runner command shown in the reproduction, using test/integ.example.test.ts and the --watch flag. Trace where the runner writes cdk.context.json or cdk.json, then verify that running it creates or updates cdk.json with the watch field without creating cdk.context.json.

Written by the indexing model from the issue text.