PA28 Flight Instructor on training flight with student reported a near miss with another aircraft in the airport traffic pattern.

Narrative

During a training flight involving pattern work I was instructed to depart ZZZ's XXR and remain in right closed traffic. I departed behind a Cessna 172 unaware that they are remaining in the pattern. During my crosswind turn to downwind; Tower advised me traffic to follow was the Cessna. However; there was also a Cessna 152 coming in from that general direction to enter in the pattern. I spotted the Cessna 152 and advised 'traffic in sight.' Tower instructed me to follow that traffic. Before my downwind to base turn; I got the instructions that I'm #2; XXR; cleared for the option. I repeated my clearance and still had the Cessna 152 in sight as they were now on final passing my right wing. I turned right base. During my turn to final everything seemed normal; however; within 2.5 miles from the approach end of the runway; ATC asked me if I had the Cessna in sight very exclamatory. I replied 'in sight.' The Cessna 152 I had in sight was departing off of the runway. However I heard the engine of another plane. Apparently; I had cut off the Cessna traffic Tower intended me to follow and we were in very close proximity with each other. I veered right and was told to go around and make right closed traffic at the departure end of the runway. The Cessna 172 I cut off was told to go around and overfly XXL. I take responsibility in the event that occurred as I did not practice what I preach with 'checking final before making the turn to final.' I'm a newly minted CFI and was preoccupied talking about descent; flaps; speeds; etc. during the pattern work. It was an embarrassing moment that showed me how crucial it is to not let teaching preoccupy your fundamental flying habits. When I reflect on the moments leading up to the near miss I think about how the radio calls all made sense. I saw a Cessna on short final; I was #2; XXR cleared for the option. However; technically during that moment I was really #3.Regardless; I spoke to the pilot I cut off on the ground and talked about what had happened. She told me that tower didn't specify which Cessna was to follow as were all in the general vicinity. She had me in sight during my mess up and was able to safely avoid contact since I was in a stable descent and was predictable. Nevertheless; I still take responsibility for not checking final during my turn from base to final.