C172 flight instructor reported on takeoff observed traffic flying incorrect pattern converging opposite direction resulted in taking evasive and an NMAC event.

Narrative

My student and I were departing ZZZ Runway XX after multiple touch and go landings in the local traffic pattern. On a previous trip around the pattern I heard a Aircraft Y report 10 miles north inbound for left traffic Runway XX. I advised the other pilot that the winds were light out of the South; that the traffic had been using Runway XX; and that there were 3 other aircraft in the pattern. On our final touch and go landing I advised ZZZ traffic that we (Aircraft X) would be departing Runway XX for a straight out departure to the South and when clear of the pattern would be proceeding northwest bound to ZZZ1. While we were on the takeoff roll I heard Aircraft Y report entering left downwind for Runway XX. They should have been no factor for us departing straight out. In the climbout I noticed another aircraft to our right pop up on our ADS-B display at our same altitude and same direction (Southbound). I immediately began looking for this other aircraft and got it in sight turning directly towards us at the same altitude in approximately a 30° bank to the left. I immediately took control of the aircraft from my student and entered an abrupt climb while advising the other traffic in the pattern that there was a Bonanza turning left base gear down for Runway XX. The other pilot came back on the radio and said he was not lost but looking for the traffic on the upwind leg. This makes no sense because if he did not have us in sight there would be no reason for him to turn in our direction. Upon landing safely at ZZZ1 I reviewed the Flight Aware data and saw that the other pilot approached the airport from the northeast; crossed over the field over the departure end Runway XX on approximately a 230° heading; descended from 2;500 feet into the left downwind for Runway XY and proceeded to turn base for Runway XY without clearing the area and preparing to land with a tailwind and against the traffic pattern in use. After the near miss the other aircraft; Aircraft Y proceeded to continue eastbound; now on a crosswind leg for Runway XX and then entered the pattern and landed on Runway XX. The correct way for this pilot to have entered the pattern according to the AIM was to proceed southbound on the east side of the airport and outside of the traffic pattern; descend to traffic pattern altitude while outside of the pattern; turn westbound; the then enter the left downwind for Runway XX. Also upon further investigation via Flight Aware; it seems like that ZZZ is the home airport for this aircraft with multiple visits in the last couple of weeks. I am trying to figure out how this other pilot became lost in the traffic pattern at his home airport and then proceeded to become a collision hazard for an aircraft that was proceeding on a straight out departure and then admit to not having the upwind traffic (us) in sight and then turn directly in the direction of the traffic he said he did not have in sight.