Forward domain resolution to AWS's namesever

By looking at the snippets I would assume that you have configured a private hosted zone (you have a private IP for the server you try to resolve) and you already have some network connectivity between your local bind and AWS (VPN or Direct Connect).

Basically, you are on the right track after your update. You need to configure BIND to act as a forwarder for the hosted zone in question. If the hosted zone is associated with your VPC in which the server resides, and if you have a proper connectivity you can forward directly to the VPC DNS server, it's address is usually the second IP of the VPC CIDR range (e.g. 192.168.0.2)

In this case you are forwarding your requests directly to AWS.

A slightly different version of the same would which was pretty common for a while is to have an EC2 instance running in the same VPC which acts as a forwarder to Route53 and then on your local machines configure the IP of the EC2 instance to be your DNS server, which in practice is the same thing you did, with the difference that BIND is running on EC2 instance within the VPC.

Nowadays the AWS recommended approach is to use the so called Route 53 Resolver with an Inbound Endpoint. Same prerequisites described above should be met - you need to have internet connectivity between on-prem and AWS, and your VPC should be associated with your hosted zone. The difference here is that instead of having to run a dedicated EC2 instance which you need to secure and administer you now have an AWS service doing that for you. In addition to that you gain some resilience and HA, as you can set up the Route53 Resolver in multiple subnets (AZs). On top of that you can create an outbound Endpoint too, which will give you the possibility to forward requests from AWS to on-prem DNS servers, and thus resolve DNS queries the other way around). You can read more about Route53 Resolver in the docs.