Flight Instructor reported a NMAC while in the pattern at a non-towered airport. The instructor stated there were three aircraft in the pattern; a fourth entered using a non standard pattern which caused the NMAC.

Narrative

Aircraft X was completing patterns at ZZZ using Runway XX. Winds were favoring XX. Aircraft Y was inbound and called over the radio that they were using Runway YY. Aircraft X said they were currently using Runway XX. Aircraft Y again said they were going to cross midfield for Runway YY. Aircraft X at this time decided to switch runways and follow Aircraft Y in for Runway YY to avoid a collision hazard.At this time Aircraft Z was about 7 to 8 nautical miles from ZZZ inbound to land and called that they would be following Aircraft Y and Aircraft X in for Runway YY by crossing midfield and entering the left downwind. When all aircraft were on the ground Aircraft X and Aircraft Z called over the radio that they were switching back to Runway XX. After this call Aircraft A called that they were 10 miles out; no direction specified; inbound for Runway YY.At this time Aircraft X departs Runway XX to continue take-off and landing practice. Aircraft X then calls crosswind for Runway XX. At this time Aircraft A calls over the radio they are 5 miles North inbound for Runway YY. At this time Aircraft Z is departing Runway XX for take-off and landing practice and calls over the radio their intention. Aircraft A then calls over the radio that they will cross Runway XX and enter the downwind for XX to follow the flow of traffic.As Aircraft Z climbs on the departure leg. Aircraft Z continued to climb through 700AGL straight out so as not to turn into Aircraft X in the downwind and to leave the pattern; depart to the North and re-enter the pattern because Aircraft Z did not have eyes on Aircraft A. The instructor in the right seat then looks right and sees Aircraft A headed on a collision course with Aircraft Z. At this time Aircraft Z and Aircraft A are at the same altitude.Aircraft A; rather than crossing mid-field as they had called over the radio; had elected to enter the pattern in the crosswind for Runway XX. Aircraft Z calls out to Aircraft A asking if they see them and Aircraft A responds that they have them in sight in the downwind. Aircraft Z reply's stating that they have the wrong aircraft in sight and that Aircraft Z is on the departure leg. Aircraft A then gets eyes on Aircraft Z and breaks to their right to avoid Aircraft Z. Aircraft Z then breaks to the left after verifying what Aircraft A's actions were. Both aircraft came close enough that the pilots of both aircraft could visually identify each other; at least 200-300 ft.A discussion between the pilots on the ground revealed that Aircraft A was inbound to attempt to enter the pattern for Runway YY at an angle in the upwind of Runway YY rather than entering 2 miles out and crossing Runway YY perpendicular to the runway; which would have put them at least 300 et above Aircraft Z as they departed and turned crosswind at 700AGL. It was also revealed that Aircraft A thought about breaking off; departing North; and re-establishing a mid-field crossing for Runway XX; but instead; elected not to and entered the pattern in the crosswind for Runway XX. The pilot of Aircraft A was aware that an entry in the crosswind of a runway rather than crossing over mid-field was not the correct or expected way to enter a traffic pattern.No-collision occurred during this incident. Thoughts on contributing factors and what could have been done differently: 1. Aircraft Y should have entered the pattern for the active runway being used by Aircraft X. The winds were light and both runways were long enough for the Aircraft Y to safely land in the given conditions. The switch between runways created some confusion. 2. Comms - the instructor in Aircraft A had trouble hearing what was happening in the traffic pattern at ZZZ. they did not possess a noise cancelling headset and stated they did not turn up the volume to better hear what was happening over the radio. Long-term and regular exposure to loud noises in the cockpit due to no noise cancelling headset usage may also contribute to hearing loss that may have played a factor. 3. Non-standard Traffic Pattern Entry - Aircraft A elected to perform non-standard pattern entries for both runways. The decision to enter the traffic pattern correctly or to break off their approach and enter correctly would have provided ample opportunities to accurately assess what was happening in the pattern; created correct separation of aircraft; and provided more options for evasive action. 4. The pilot of Aircraft A miss-identified Aircraft X as the aircraft calling over the comms and thought there was only one aircraft in the pattern. Fixation inside the cockpit rather than eyes outside may have contributed as well. 5. Not using all resources available - Aircraft A was equipped with a Stratus ADS-B In receiver that could have been utilized to have better situational awareness of local traffic on the instructors IPAD. 6. Tail of Aircraft Z close to Runway YY in name; expectation bias - Aircraft A may have been listening for key words and heard YY; assuming it meant Runway YY was in use; elected to approach for Runway YY not realizing Aircraft Z was in the traffic pattern.