C172 Flight Instructor reported having to execute an evasive maneuver during an NMAC event while on visual approach as the other aircraft misidentified traffic; resulting in a loss of separation.

Narrative

My student and I had just departed Runway XXR to do some training in the traffic pattern in ZZZ. We were instructed to extend our departure leg to allow aircraft on the parallel to turn crosswind before us. After that; Aircraft Z turned crosswind. We were given clearance to turn crosswind about another mile beyond them to join left traffic for XXL to start pattern work. Aircraft Y behind me was given instruction to extend upwind and to look for Aircraft X traffic on the crosswind leg - my student and I - and to report them in sight. He reported that he had the Aircraft Z traffic in sight and Tower gave them instructions to turn crosswind north of the Aircraft X traffic. Aircraft Y appeared to have seen the Aircraft Z ahead of us that was already on downwind but had not seen us turning crosswind to downwind. On downwind; I heard the situation play out on the radio and became aware of a potential hazard as we approached the normal crosswind leg. I saw an aircraft on the ADS-B screen on our transponder showing it approaching from our left and climbing with 200 - 100 feet below. I grabbed controls of the aircraft as Tower began advising Aircraft Y of Aircraft X traffic ahead; and told him to descend 200 feet immediately. I saw the aircraft approaching us directly from the left and did a climbing left-hand chandelle to avoid the converging traffic that appeared to still be climbing. I am not certain that they saw us in time to begin a descent; which is why I did not initiate a right-hand turn as stated in Part 91 for right-of-way rules. The aircraft departed the airspace to the west and we continued flying out traffic pattern with no action towards the other aircraft from ATC.