C172 flight instructor reported based on traffic pattern reports; pilot started take-off and aborted as traffic was observed opposite direction on final; resulted in a critical ground conflict.

Narrative

On Day 0; around XA:30PM; I was conducting a routine training flight with my student. The lesson for the day called for short/soft field take-off/landing practice; and we opted to fly south to nearby ZZZ from our home airport; ZZZ1 to practice. After a few uneventful laps in the pattern using Runway XX with one other aircraft in the pattern with us (wind favoring XX as well as YR/ZZL being closed at the time) a Piper Cherokee; joined the pattern to make 3 total in the pattern (including myself). The Cherokee appears to be from a local flight club at the airport; and taxied to Runway XX shortly after we landed on our 3rd lap. As we pulled off; the Cherokee took off from XX uneventfully and we began our taxi back to Runway XX. As we taxied; ADS-B data showed the Cherokee turning right and joining a pattern for YR; the closed runway. Shortly before we reached XX; I asked the Cherokee their position; to which they did not answer. Shortly after; they reported right base for Runway XX. Holding short of XX and ready to go after clearing the area of traffic; my student commenced the take-off; believing we had more than enough time with the aircraft just beginning a right base. ADS-B showed they were still around the YR upwind pattern area; and I believed the data to be wrong as it sometimes glitches; and the runway was closed with big Xs on both ends; as well as the pilot reported a right base for the runway in use; XX. Shortly after full power was applied; to my horror; I look up and see the Cherokee on short final for Runway AA; the opposite runway. I immediately aborted the take-off; and urgently called for the wrong way traffic to go around. I had nowhere to exit the runway due to my position on XX between the taxiways but was ready to pull the airplane off the side of the runway to avoid the collision. Thankfully the traffic went around; and shortly after said 'My Bad'; and we exited XX. The other aircraft in the pattern for XX with us was also astonished by the actions of the Cherokee. I would estimate the near miss to be around 300-400 ft. After almost causing a collision; the pilot of the Cherokee began erratic flight just north of the airport at a low altitude; so much so other aircraft asked if they were ok. They also announced they were joining the pattern again following the incident but would fly away back north of the airport. The pilot of the Cherokee operated the aircraft in an extremely unsafe manner and almost caused a wrong way collision. As an instructor; the incident was a good 'Double Check' learning experience. Even though the pilot reported one thing; other data had a discrepancy on the report.