Enabling `BinaryMediaTypes` on `AWS::Serverless::Api` Base64-encodes request payload body

### Description

When enabling `AWS::ApiGateway::RestApi`'s [`BinaryMediaTypes`](https://docs.aws.amazon.com/serverless-application-model/latest/developerguide/sam-resource-api.html#sam-api-binarymediatypes) for a particular type (such as `application/json`), this causes the _request's_ body to have been Base64-encoded when received, as well as the expected behaviour of allowing the API to return binary data as a Base64-encoded body within the response payload.

This is unexpected behaviour given the documentation (see below for why), or it could be a bug. Getting clarification on which it is is critical to the use of these resources in AWS; if a documentation issue, the published info should be clearly explained.

The docs on `AWS::ApiGateway::RestApi`'s [`BinaryMediaTypes`](https://docs.aws.amazon.com/serverless-application-model/latest/developerguide/sam-resource-api.html#sam-api-binarymediatypes) suggests this only affects the response (emphasis mine):

> List of MIME types that your API **could return**. Use this to enable binary support for APIs. Use ~1 instead of / in the mime types.
>
> AWS CloudFormation compatibility: This property is similar to the [BinaryMediaTypes](https://docs.aws.amazon.com/AWSCloudFormation/latest/UserGuide/aws-resource-apigateway-restapi.html#cfn-apigateway-restapi-binarymediatypes) property of an AWS::ApiGateway::RestApi resource. The list of BinaryMediaTypes is added to both the AWS CloudFormation resource and the OpenAPI document.

The docs above appear outdated as the `Use ~1 instead of / in the mime types.` reference no long applies - using `application/json` works fine via SAM locally and when deployed to production AWS. Online resources such as [this StackOverflow answer](https://stackoverflow.com/questions/59615948/how-can-one-return-binary-content-via-aws-lambda-through-api-gateway-and-cloudfr/60977620#60977620) suggest confusion and suggest the AWS docs were wrong and that the `AWS::ApiGateway::RestApi` has since been fixed up.

On the linked [`AWS::ApiGateway::RestApi`](https://docs.aws.amazon.com/AWSCloudFormation/latest/UserGuide/aws-resource-apigateway-restapi.html#cfn-apigateway-restapi-binarymediatypes), the docs are unclear:

> `BinaryMediaTypes`: The list of binary media types supported by the RestApi. By default, the RestApi supports only UTF-8-encoded text payloads.

### Steps to reproduce

```yaml
Resources:
MyAPI:
Type: AWS::Serverless::Api
Properties:
StageName: prod
BinaryMediaTypes: ["application/json"]
```

### Observed result

1. Deploy a SAM template such as that above, with a corresponding Lambda function
2. Observe that in an AWS Lambda function, the `isBase64Encoded` property on the request is `true` and the `body` is encoded with Base64.

### Expected result

1. Establish if the behaviour of Base64-encoding the request payload body when `BinaryMediaTypes` is enabled is expected.
1. Update the documentation for https://docs.aws.amazon.com/serverless-application-model/latest/developerguide/sam-resource-api.html#sam-api-binarymediatypes accordingly:

1. Indicate that the `BinaryMediaTypes` option will Base64-encode the request body and require that the response body be Base64-encoded.
1. Remove or correct the `Use ~1 instead of / in the mime types.` reference.

1. Update the document for https://docs.aws.amazon.com/AWSCloudFormation/latest/UserGuide/aws-resource-apigateway-restapi.html#cfn-apigateway-restapi-binarymediatypes accordingly to indicate the same impact above.

If the behaviour is _not_ correct, then the resolution would be to fix how SAM and AWS operate.

### Additional environment details

1. OS: Ubuntu 24.04
2. If using the [SAM CLI](https://github.com/aws/aws-sam-cli), `sam --version`: `SAM CLI, version 1.142.1`
3. AWS region: `us-east-1`