Add operator or API run_type's to dag runs

### Description

Any dag runs created by the REST API or trigger dag operator will appear as a `manual` dag run. That can be confusing. Instead we could add two new `run_type`s for `API triggered` `Operator triggered`. Open to better names.

For the API, a user should also be allowed to specify the `run_type` because sometimes it still is a manual button to click.

We may need to add checks to make sure it is a valid run_type.

### Use case/motivation

_No response_

### Related issues

_No response_

### Are you willing to submit a PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

Start by tracing DAG run creation through the REST API and the trigger DAG operator, then inspect how run_type values are represented and validated. The work is done when API- and operator-triggered runs can be distinguished from manual runs and the API accepts only valid run_type values, including an explicitly chosen type.

Written by the indexing model from the issue text.