When executing a testvia the AWS API Gateway console, I see that the output response from the Lambda function is correctly transformed:
{
\"type\" : \"\",
The difference between Test Invoke on API Gateway Console and actual Invoke with API Gateway is the test invoke is always assuming the content-type is "application/json" and accept "application/json". You might want to pass "application/json" with the header accept when you invoke via Postman.
Before the changes are externally visible, you need to Deploy the API !