Instructor pilot reported a NMAC with an aircraft while in the airport traffic pattern.

Narrative

Flying Left patterns for Runway XX into ZZZ with a new student; we were on left downwind with one airplane ahead of us on base turning final. Was also aware of traffic from the east coming in. Standard call outs at this field are on base when flying patterns; also wasn't told to report anything. Pattern prior I called on left downwind. Still on downwind I became aware of the traffic coming in flying almost parallel to our course. At this time a Aircraft Y calls in; ATC misses the call; so Aircraft Y has to repeat. Unable to report downwind due to this and plan on reporting base. Still on downwind/extended downwind at this point with traffic we are following on final not yet abeam and another airplane parallel to our course at the same altitude. Before we are able to turn base due to the traffic on final an ADS-B traffic warning goes off for the traffic inbound and parallel. At this time; I believe the traffic is either on final or entering a close left base to final. With incoming traffic and another plane on final to my left practicing touch and goes. I notify ATC of the alert and using my captain's authority; execute the standard right turn for head-on traffic avoidance; believing that will take me well clear of the traffic. Traffic then mirrors my turn to the left; following my course. I begin executing a climb with the traffic nearing as I noticed I was higher; and we pass within 200 ft. of each other. I notify ATC again of what happened. Not sure if opposing traffic ever realized I was there; aircraft was ADS-B equipped; but didn't respond at all to my call nor acknowledge it. May have also been making a left turn to intercept the extended base he was initially given; as what he was flying before his correction was more of a final or shallow base entry. Afterwards; a little shaken; I may have misheard a call telling me to enter a right downwind for Runway XX. I ask for clarification; stating we are to the south of the field and a left downwind would be best for Runway XX. Get told to enter left downwind for XX and proceed to follow Aircraft Y in who called earlier in.