minor: Unify and adjust API Gateway stage name

[whummer]

When preparing for a customer demo today, we've noticed that the API Gateway URL does not work for the seed script.

$ bin/seed.sh
...
[2025-04-09 11:48:47] Creating Comic Book Quiz...
[2025-04-09 11:48:47] ERROR: Failed to create quiz: {"message": "The API id 'egjlwwwc4s' does not correspond to a deployed API Gateway API"}
...

Got some help from @bentsku today 🙌 who found out that the stage reference needs to be adjusted in the API Gateway URL, to be compatible with AWS (and the latest provider implementation in LS).

[github-actions]

🚀 Preview for this PR: https://ls-4d7tkepavuers.sandbox.localstack.cloud/cloudfront/aed84b3b/

[bentsku]

Quick context as to why it failed: https://docs.aws.amazon.com/cdk/api/v2/docs/aws-cdk-lib.aws_apigateway.RestApi.html#deploy

To unblock the demo, we ran a awslocal apigateway get-stages --rest-api-id <id> to see the existing deployed stages and saw the prod one.

I've just looked at the docs: when deploying with CDK, it will auto deploy your REST API, apparently with the prod stage name.

We could also set some parameters over the RestApi construct with deploy=True and deployOptions and https://docs.aws.amazon.com/cdk/api/v2/docs/aws-cdk-lib.aws_apigateway.StageOptions.html, so we could specify the stageName.

Anyway, just wanted to understand myself why it suddenly deployed on the prod stage, and give context back 😄