API Gateway V2, creating a Private REST API

Using the old resource “aws_api_gateway_rest_api”, you could link to a private vpc endpoint using a configuration similar to the following.

resource "aws_api_gateway_rest_api" "example" {
  // The bulk of this resource's body was omitted for brevity

  endpoint_configuration {
    types            = ["PRIVATE"]
    vpc_endpoint_ids = [aws_vpc_endpoint.example[0].id]

Similarly in the AWS console you can select “Rest API Private” when creating a new API Gateway resource:

Using the new API Gateway V2 resources, how do I configure my API Gateway in a similar way?
All the documentation I could find is referencing the old (non V2 resources), or non-terraform infrastructure management (AWS CLI, AWS Console, etc.)

It looks like the aws_apigatewayv2_api (apigatewayv2) is different than aws_api_gateway_rest_api (apigateway).
The apigatewayv2 would seem to only be able to create HTTP APIs, which do not currently support Private Endpoint types.
REST API does support Private Endpoint types.
However, REST API is something exclusive to apigateway , not apigatewayv2 .
The AWS documentation does a bad job explaining this, the console tries to disguise it, and the differences are dangerously important.