C172 pilot reported performing evasive action after encountering a NMAC at a non-towered airport. The pilot was focused on maintaining separation and making radio calls with one aircraft and lost situational awareness of another aircraft that was turning to join downwind.

Narrative

I went to go fly for my CFII checkride with [a] DPE. During this flight; we shot the VOR; circle-to-land at ZZZ; and departed on the downwind for the GPS XX into ZZZ. After we did a touch-and-go; I reported over the ZZZ CTAF that I was departing on the downwind to set up for the GPS XX approach. While I was turning my downwind; we received a traffic alert for another aircraft in the pattern. While we were focused on the traffic alert; we received another one; and out of the corner of my eye; I see a plane virtually at the same altitude flying straight for us. I immediately pulled power and started a heavy descent to avoid this traffic. Upon checking our ADS-B In on the MFD (Multi-function Display); we saw a plane pretty much right on our tail. After getting home; I looked at the ADS-B exchange flight log. We had 200 - 300 ft. of vertical separation; and somewhere between 100 - 200 ft. of lateral/horizontal separation. During this flight; we had constant thermals that would have us gain 400 ft. in altitude. I believe that there were many factors that contributed to this loss of separation. While flying in the busy traffic pattern; I was focused on maintaining separation; and making radio calls. When I got the first traffic alert; I got focused on maintaining separation with that aircraft and had a slight loss of situational awareness with the other plane that was inbound in a 45-degree entry into the opposite runway into the pattern. Upon realizing that the other aircraft was headed straight towards me; I saw the plane was trying to do a 270-degree turn to join downwind without paying any attention. When I was initially turning downwind; they had called us to see if we were departing. I had told them that I was departing on the downwind; but I believe that they misunderstood that I was trying to depart on the crosswind and they proceeded to do a 270-degree turn without looking outside. This experience had taught me that any loss of situational awareness can be potentially fatal.