What do statements of interest, teaching, and research mean when applying for an academic job?
That is a rather confusing request, actually. However, what I think the job offer is looking for in a "statement of interests" is a description of the research areas you are interested in, while the "statement of research" would be a listing of projects you'd plan to start if you were hired. 

However, in many job postings, those two items would potentially be conflated into a single statement. So I'd recommend that you contact the department. I'd recommend doing this by phone, if possible—that way, you don't have to give as much identifying information.