Are AWS CodeStar events considered Management or Data events?

I'm building a CodeStar project and want to create a CloudTrail trail to record its events - more specifically, the events found in the documentation here. On the page where I can create a trail, I am unable to select anything related to CodeStar under the "Data events" section. However, I am able to search for CodeStar related events under the "Event history" section.

Does this mean that the events listed in the documentation are Management events?

Yes, the events mentioned in the document you shared are Management events as they are control plane operations. Data events are only related to S3 and Lambda.