In al the tutorials about Kubernetes cluster I have read I didn't see that they mention to 2 load balancers, but only one for the ingress pods.
However, in a proper production environment, should's we have 2 different load balancers?
For all production environments its advised to have load balancer for API Server. This is the first step as part of K8S HA mode creation. More details are in k8s documentation
You are correct for this also it’s definitely required to handle external traffic. All the ingress services are created of LoadBalancer Type in their implementations.