C172 student pilot flying solo reported NMAC in airport traffic pattern.

Narrative

The day after my dual air work practice; I did my first solo air work practice in the north practice area. Going out to the practice area and practicing my airwork was not a problem. When it was time to return to ZZZ; I contacted ZZZ Tower; and they told me to enter a left downwind to land on Runway XL. When I was on midfield; left downwind; Tower told me to extend my downwind and said that he will call my turn to base. Tower also notified me that there was a TBM who was entering a right downwind (just for my information). At that time; I did not know what a TBM was; and I said that the Tower will call my base. Tower reached out to me again and asked if I have found the TBM and said; 'He's in a right base;' and told me to report him if I had the plane in sight. I misheard Tower and thought that he said; 'please enter a right base' instead of 'he's in a right base.' In my mind; I thought that he wanted me to turn base ONCE I have found the TBM. As I was looking for it; I saw a plane on upwind for Runway XL and thought that that was the TBM. It turned out that it was a Skyhawk. Because I thought I saw the TBM on upwind; I told Tower that I had the traffic in sight and started my turn to base (I thought I had clearance to turn base). As I was turning to base; Tower asked again whether I had the TBM in sight; and I said yes. While I was turning base; the rapidly descending TBM overshot final by at least 3/4 of a mile; facing the plane that was behind me on the downwind. Tower told that plane to turn to the north to give more spacing for the TBM. While the TBM was overshooting; Tower asked if the TBM had the Skyhawk traffic in sight; referring to me. The TBM said that they did have the traffic in sight. As I was turning to final; I felt like something was not right. During that time; Tower was busy communicating with other people who were also in the pattern. When no one was talking over the radio; I asked Tower (just to make sure) if I had clearance to land. Tower said; 'negative.' I started looking for other traffic around me and saw the TBM flying a couple hundreds of ft. above me; who was also on final. My instinct was to get as far away as I could from the TBM; so I started banking to the right. Without realizing; I crossed Runway XXL's path and was on right downwind for Runway XL; flying the opposite way.When I realized I was on the wrong path; I quickly climb to above 3;000 feet MSL. Tower later called me and asked if I saw the TBM who was on base earlier; and I said no. Tower told me to make a left closed traffic and to start my left turn to crosswind. At that time; I did not know what he meant by a 'left closed traffic.' I thought he wanted me to just turn left to crosswind; so I did. As I was on crosswind; I asked if I could turn left to enter downwind again; but Tower did not reply (Tower was really confused with my request). I thought; 'Since Tower hasn't given me permission to turn left to downwind; I'll continue flying straight until he tells me to turn. And if he doesn't; I'll continue flying and exit the pattern.' I thought it was also a good idea to exit the pattern since he did not tell me to turn left to downwind; so I flew straight; heading to the north practice area. As I was exiting the pattern; Tower reached out to me and asked what I was doing. I told him that I just wanted to enter left downwind for XL. Tower told me to turn around and to re-enter the traffic pattern. I entered left downwind for Runway XL and landed safely.