C-172 Pilot in airport traffic pattern reports NMAC with another aircraft also in airport traffic pattern.

Narrative

I was flying Aircraft X on the day of the close call; which occurred around XA:30 local time. The weather at the time was VFR; with good visibility and broken clouds at around 11;000 ft. AGL (per the METAR from around the time). Surface winds were from the east/southeast at 5-10 kts.; but it felt as if there was a wind shear layer at around 500 ft. MSL. At the time; I was about one hour and 20 minutes into pattern work at (ZZZ). Runway XX was in use; and I was sharing the pattern with a high-wing Cessna (I forget the exact model) - I'll refer to it as 'Cessna 2' in this report for simplicity.Leading up to the close call; I was monitoring ZZZ traffic on CTAF and communicating intentions per standard operating procedures. I departed Runway XX after declaring intentions to stay in the pattern; making right closed traffic; and the only other aircraft I heard on CTAF at that time was Cessna 2; which had just taken off and was making right closed traffic. At approximately 700 ft. MSL; I initiated a climbing right turn to enter the crosswind leg of the traffic pattern and simultaneously announced on CTAF that I was on right crosswind for Runway XX. After completing the turn; and while on the crosswind leg; I noticed an aircraft that appeared to be heading almost directly toward me at a slightly higher altitude; but was unable to determine the exact distance away. As I prepared to turn right to join the right downwind leg; I heard a call on CTAF from Aircraft Y indicating that it was on course to intercept the right downwind leg for Runway XX at a 45-degree angle. The call did not indicate any distance from the traffic pattern; and this was the first call I heard from Aircraft Y on CTAF that day. I assumed that the aircraft I saw heading approximately toward me was Aircraft Y based on this call.As I completed my turn to intercept the right downwind leg; it became clear that Aircraft Y was quite close to the traffic pattern (probably less than one mile); was at a similar altitude to me (i.e.; approximately 1000 ft. MSL); and was on a converging course with me to the right downwind leg for Runway 10. It also became clear that unless immediate action was taken; there was a high chance that I would collide with Aircraft Y on the downwind leg. I indicated on CTAF that I had turned right downwind for Runway XX and asked Aircraft Y to alter course to enter the traffic pattern behind me to avoid collision. Aircraft Y acknowledged and complied; passing behind me at what appeared to be a distance of several hundred ft. away.After this point; I continued on the right downwind leg and announced intentions to execute a simulated engine out landing for Runway XX. Shortly after this; I noticed Cessna 2 turning from right base to final for Runway XX and heard a call on CTAF from Cessna 2 indicating as such. I also heard a subsequent call on CTAF from Aircraft Y advising me of the traffic on final and urging caution; I proceeded to make a normal landing to a full stop behind Cessna 2. After one more uneventful circuit in the pattern; I taxied to the fuel pumps; where I waited behind Aircraft Y; which was being refueled.While at the fuel pumps; I introduced myself to the Instructor onboard Aircraft Y (which was on a student training flight) and asked to speak to him about what had happened. During the conversation; the Instructor indicated that Aircraft Y had not made any initial call prior to their call indicating that they were on a 45-degree course to the right downwind leg for Runway XX. The student pilot later clarified that they had indeed made an initial call; but it was on the incorrect frequency; which is why I didn't hear any such call. The Instructor also indicated that Aircraft Y had heard my call indicating that I had turned to join the right downwind leg of the traffic pattern as they approached the pattern.I've given quite a bit of thought to my decision making in the situation as well as factors related to my flying; the aircraft; the environment (including Aircraft Y); and external pressures that led to the outcome that occurred. I would make the following observations:1. The first call I heard on the CTAF from Aircraft Y was while the aircraft was on its 45-degree course to the right downwind leg for Runway XX. The fact that this call came in so shortly before Aircraft Y joined such leg as well as the lack of any call from Aircraft Y while approaching the area resulted in me being caught off guard that traffic was approaching so close to my position. Frankly; I was surprised and unprepared for how quickly the traffic converged on my position after hearing the initial call from Aircraft Y. This was a particularly good lesson for me to remain vigilant for traffic that may not necessarily be communicating on/monitoring CTAF or may not necessarily have a good handle on the traffic operating in the pattern.2. When it became clear that Aircraft Y was attempting to join the pattern in a manner that could have resulted in a collision course with my aircraft; I could have been more proactive about adjusting course; rather than urging Aircraft Y to adjust course. Part of my reluctance was likely the fact that I was already in the pattern; believed that I had the right of way; and likely had some level of fixation on completing the circuit in the pattern as planned. Furthermore; I had a feeling of being boxed in (due to the nearby surface Class B Airspace (immediately ahead of the departure end of Runway XX); the overlying Class B Airspace shelf (at 1500 ft. MSL); and the proximity to the ground. I could have considered an eastbound turn or an immediate climb under my authority under 14 CFR 91.3 even if it meant entering the Class B Airspace.3. The fact that the Aircraft Y seemed to be on almost a direct heading to me made it very difficult to judge the actual distance of the aircraft. In the future; when seeing an aircraft heading directly or nearly directly for me; I would probably be more proactive (regardless of my estimation of the distance of the aircraft) in considering evasive action or verifying distance of the aircraft through other means (e.g.; ADS-B on the GPS map display).4. For the previous 20 minutes or so; I had been in the pattern with Cessna 2 and was focused on avoiding this particular aircraft. This dynamic likely had me focused on the status quo; which involved the two of us sharing the pattern; rather than focusing on other aircraft that could potentially join the pattern. 5. I was not actively monitoring ADS-B traffic on the GPS map display at the time. This was probably compounded by the fact that the GPS aviation database was not available for the flight; potentially leading me to have subconsciously discounted the usefulness of looking at the map display at all during the flight (although the display was providing information about traffic).6. While well within my capabilities; the wind conditions were a bit challenging; particularly above 500 ft. MSL; where there was consistent light turbulence that required frequent control inputs to fly the desired course and climb profile. This; combined with the fact that I had not flown [in a while]; probably resulted in me spending a disproportionate amount of attention on controlling the aircraft that compromised my ability to look for traffic in the area.