GA pilot reported an NMAC event in the traffic pattern after an ATC trainee mistakenly cleared two aircraft to land on same runway simultaneously. ATC issued an immediate 'left 360 for traffic separation'.

Narrative

I approached APC from west; and upon contact with APC Tower; I was advised to make right traffic for 19R; and report midfield on downwind. I read back those instructions. Upon turning to downwind - before I was at mid-field; APC Tower used my call sign and said 'cleared to land 19R ' and advised there was traffic to the south. I read back all instructions. As I was turning to a right base leg; I heard Tower use my call sign but the call was not clear. As I continued to turn onto a right base I asked for a repeat of the call. ATC advised me to begin an immediate left 360 for traffic separation. I immediately executed a left turn while maintaining about 1;100 feet altitude. At about the same time I heard Tower clear another aircraft for landing on 19R. The inbound aircraft was a south bound business jet. ATC asked both aircraft if we were had a visual. The jet saw my aircraft and I saw the jet as it passed 400 to 600 feet beneath me. I continued with the turn and ATC cleared me to land on 19L. After landing; ATC asked me to call Tower. In that call ATC asked why I had turned on a base leg before executing the left 360 as requested. I advised that I was cleared to land on 19R and already started the base turn before advised to start the turn. ATC advised they would review radio transcripts. Tower mentioned that ATC personnel were in training that day. A Tower Controller called my cell phone a few hours later and advised that a transcript review indicated that I was cleared to land on 19R prior the execution of the base leg turn. But [the Controller] indicated my radio was 'fuzzy' and implied my request for a repeat of the instruction for a 360 created some difficulty.At no time was I advised about inbound traffic from the North; and I have to conclude that when I was making the left 360; Tower thought I was still on a north bound downwind leg.