General aviation Flight Instructor reported a near miss with another aircraft in the traffic pattern during a training flight. The tower instructed the other aircraft to perform a maneuver to separate the traffic and changed the Instructor pilots landing sequence to provide more separation in the traffic pattern.

Narrative

During an instructional flight; after performing a touch and go; climbing in the pattern; we turned left crosswind. After completing the turn; the student and I visually identified traffic ahead and to our right; but very close. Due to the closeness of the two; I directed the student to turn downwind which kept lateral separation from the other aircraft; but changed the sequence of our arrival in the pattern. I contacted the tower; and they told the other (outside) aircraft to perform a 360. After listening to the tower recording; and viewing the ADS-B data; I'm aware of several factors and mistakes that led to the loss of separation.My focus was on evaluating my student's performance. While I was observing traffic visually and using an ADS-B connected iPad; I lost sight and awareness of the aircraft in the pattern ahead of us. Being seated on the right side of the aircraft; and making a left climbing turn; it was not possible for me to see the other aircraft until we completed the crosswind turn. I am now aware that the other aircraft extended upwind; possibly out of confusion based on what I heard in replaying the audio. Had ATC instructed them to extend upwind; I would have been alerted to the need for us to delay our turn. I incorrectly assumed that they were flying a normal pattern; and we did not need to delay our turn to crosswind. I visually checked to our left before making the turn; but the other aircraft was farther ahead; and not visible to me. Once we turned crosswind; and I identified the other aircraft; I made the decision to turn inside of them. At the moment; it seemed that trying to maneuver behind them would have been a more difficult maneuver. Ideally; we should not have turned crosswind; but given that we did; I can also see that returning to upwind; or just making a turn to the right would have kept us in the proper sequence. I'm still concerned that making a right turn might have brought us closer to the other aircraft which would have created a greater hazard.