PA-28 Instructor with trainee reported a NMAC at a non-towered airport when the other aircraft made confusing position reports and then appeared in the traffic pattern 300 feet above his aircraft.

Narrative

When entering the airport traffic pattern for ZZZ from the east; I (instructor on board) called standard distance and direction from traffic pattern. Advised traffic we were east of the field on a 45 entry for the left downwind; and continued making calls as advised in the AIM (Aeronautical Information Manual). RV traffic called with unclear and confused sounding position reports stating only that he was RV traffic and crossing midfield for the left downwind runway XX. Given the information provided the student and I were scanning across the airport for traffic approaching from the west to cross the midfield and enter our downwind as stated. We called established at midfield as the RV came from behind us contrary to his position report and crossed overhead with minimal separation; visually within 300 ft. The aircraft banked to his right after crossing over and did not make a call. I called him on CTAF that he got too close to us and asked his intentions. He was combative on frequency and immediately left the area to the east. We attempted to continue with the training flight; completing 1 touch and go and one full stop in the pattern before deciding to come into the FBO and file a report while it was still fresh in our minds.To summarize: We called at least 6 times without full report 'ZZZ traffic Aircraft X X miles to the EAST setting up for/established on the 45 for left downwind runway XX.' We had already called traffic in the pattern in sight and informed we would follow behind. The offending traffic made incorrect and confusing calls including nothing but the aircraft type; tail number; and 'will cross midfield for [runway] XX.' Pilot was combative when contacted and left the area promptly. This issue could be easily resolved with proper emphasis on position reporting and retraining. I believe complacency was the primary human factor in this incident. As a pilot and instructor the actions I could have taken to avoid or correct the situation that I will put more emphasis on in the future: Made contact with the pilot and asked for clarification on his confusing/ unclear calls. Asked to discuss on the ground instead of on frequency; this was an unprofessional argument to have on FREQ as an instructor. I have talked to the student about how to handle this situation better than I did should he encounter something similar in the future.