While terms like direct and indirect can be classified within a broader category (object, governance etc.), I am not sure how to classify direct and indirect speech. I thought it could be just speech, but found no evidence of that. What is the one-up here?
Direct and indirect speech are subsumed under the term reported speech. Here is an extract from the entry on reported speech from the Oxford Concise Dictionary of Linguistics (p343):
Reported speech: One utterance as reported by another ... . Strictly including cases of both indirect speech and direct speech; often, however, used of indirect speech specifically.
In language teaching in particular reported speech tends to be synonymous with indirect speech only.