… defining authorization type in method or root api (#30822)

### Issue # (if applicable)

Closes #30444

### Reason for this change

The original PR caused a breaking change, we can't rollback because it
was released in v2.142.0 and it fixes customers issues (partially).
Simply doing a revert will be breaking for those customers again.

### Description of changes

Identified the root cause and we should use `AuthorizationType` instead
of `AuthorizationTypeOption`. `AuthorizationType` defaults to find the
authorization type from the authorizer, falling back to use the auth
type defined in the `Method` construct's options property and falling
back to `None`.

`AuthorizationTypeOptions` on the other hand tries to find the auth type
from `Method` construct's options property which can be None because
it's optional.

### Description of how you validated changes

New unit tests covering the changes and new integration tests covering
it.

### Checklist
- [ ] My code adheres to the [CONTRIBUTING
GUIDE](https://github.com/aws/aws-cdk/blob/main/CONTRIBUTING.md) and
[DESIGN
GUIDELINES](https://github.com/aws/aws-cdk/blob/main/docs/DESIGN_GUIDELINES.md)

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*By submitting this pull request, I confirm that my contribution is made
under the terms of the Apache-2.0 license*

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