How to correctly formate the response for an AWS Lambda API?

I'm trying to understand how to correctly format the JSON that is sent back from my AWS rest API. I am using Api-Gateway to create an API that return the result of a Lambda.

I am used to API responses in the following format:

{
    "name" : "John",
    "age" : 34

}

But, in the AWS example for integrating lambda and api-gateway, they said to use the following response format:

// The output from a Lambda proxy integration must be 
// in the following JSON object. The 'headers' property 
// is for custom response headers in addition to standard 
// ones. The 'body' property  must be a JSON string. For 
// base64-encoded payload, you must also set the 'isBase64Encoded'
// property to 'true'.
let response = {
    statusCode: responseCode,
    headers: {
        "x-custom-header" : "my custom header value"
    },
    body: JSON.stringify(responseBody)
};
return response;

Is there a right way to do this, or can you just return whatever response style you like?