C172 pilot conducting solo training maneuvers reported the engine quit during a maneuver due to the reporter using the wrong air/fuel mixture for their altitude and type aircraft.

Narrative

As part of instrument rating and commercial certificate preparation and learning; I conducted a local solo flight; using a flight school owned aircraft; with the intention to practice commercial maneuvers and practice holds at and or near ZZZZZ waypoint. The flight was from and to ZZZ airport. I have chosen this waypoint due to a prominent visual landmark (a pump station); and a mayor highway that could be used for emergency landing; plus a nearby dirt/grass private runway; ZZZ1 Airport. I have also chosen to conduct these maneuvers at altitudes between 6;000 to 10;000 feet MSL with the intention to create plenty of separation between my aircraft and heavy training traffic commonly found around the area at lower altitudes; and present during this flight.My experience with this aircraft (Cessna Skyhawk C-172 S; 6 pack) at the time was limited; having significantly more time and knowledge on Piper PA 28-121 (Piper Pilot 100i with Garmin G3X. I started the flight by first being dispatched by school personnel. I had shown my required documents; answered questions regarding IMSAFE checklist; plan for the flight; weather briefing; legality of my navigation charts and documents; weight and balance and performance calculations. The only thing not asked during the process and important to the event was the planned altitudes which I already had decided on; and would have disclosed upon being asked.After taking off from ZZZ airport I initiated a climb up to 2;600 MSL due to the presence of clouds; staying under said clouds as required under VFR rules; until was able to resume a climb up to 9;000 msl; reaching altitude roughly around the area planned to conduct the maneuvers. As indicated on the aircraft climb and cruise checklists; I had configured the aircraft and have adjusted the air/fuel mixture upon leveling off under the cloud ceiling. Once clear of clouds; I resumed the climb in the direction to ZZZZZ; still leaned. Once reached desired altitude; I performed a dual VOR check using ZZZ VOR; then proceeded to hold at ZZZZZ; announcing intentions overa frequency; a frequency commonly used in the area for air to air communications and advisories; dominantly at this particular location; and a common frequency used between flight school aircraft.The weather throughout the flight was as brief and expected; being clear sky around my planned maneuvering area for more than enough distance. Expected winds and temperatures; not particularly turbulent or uncomfortable conditions. Once done holding for around 4-6 laps; I set the air-fuel mixture to full rich in preparation for the rest of the maneuvers. The reason for this is that up to this point; I have been instructed by all my CFI's to set the mixture to full rich before performing these type of maneuvers. Unfortunately; I was not fully aware that the POH indicates to lean the mixture for all operations over 3;000 MSL. I had the knowledge that the engine at such altitudes performs better with a leaner mixture; but wrongfully opted to follow what I have been told multiple times to do by my instructors. The instructions to set the mixture to full rich has always been given when at lower altitudes (1;000-5;000 MSL) but was never told it should only apply to low altitudes; as they have clearly never expected a student to choose higher altitudes to perform these maneuvers.I began practicing numerous maneuvers including parabolic climbs and descents; steep turns; chandelles; lazy eights; power on/off stalls; and others; occasionally announcing my movements over the frequency and looking for traffic by sight; as I did not had ADS-B in information available on the plane or on my EFB. Can't precisely recall the order; however; somewhere in between maneuvers I began a slow flight; testing the reaction of the controls at slow speed and high angle of attack; and eventually slowly entering into a power off stall maneuver as means to identify the aircraft stall indicated speed and following reactions to the stall condition. At roughly 8;400 MSL; as I was already near stall horn airspeed during slow flight; it only took a minor throttle movement to reach full idle; and was necessary to slowly add pitch up input to reach fluttering; then more pitch to reach a full stall condition. It took a fair amount of time between reducing speed to slow flight and the beginning of the stall; potentially giving the engine enough time to cool down excessively. In almost perfect synchrony; as the plane started to nose down during stall condition; the engine had a sudden drop of RPM and the propeller seemed to have come to a complete stop. Engine sound had mostly if not fully gone silent. I naturally and almost intuitively pitched down some to recover from the stall; stopping at a slight pitch down attitude. After a second or 2 just processing in my mind what just happened; I quickly assumed the engine stopped due to being too cold or for lacking air. I did not check the engine indications; instead; I quickly reacted by applying half throttle and at the same time; entering a descent maneuver to the left; in hope to get the propeller windmilling and restart the engine without using electricity. Given the altitude and proximity of suitable landing sites; this altitude tradeoff seemed plausible.Once reached fair amount of indicated airspeed in a matter of 3-5 seconds (likely aiming just below the start of yellow line on the speed indicator); I did not notice enough or no rotation at all from the prop. I turned the ignition key briefly (less than a second) to force the prop to rotate; and did start rotating; however; there was no clear immediate indication of the engine actually restarting (lack of loud engine sound; prop not rotating as fast as it would with the engine on). Now the prop turning at a slow speed and engine generating some sound; I adjusted the air/fuel mixture roughly to 50%; did not notice any immediate improvement; and quickly after; turned the ignition briefly once more; this time noticeably restarting the engine; prompting me to quickly adjust the throttle and mixture to a state where the engine ran comfortably; before setting full throttle and recover from the descent maneuver; returning in a timely manner to a higher altitude.The aircraft never came close to reach never exceed speed; or critical structural loads. The time between presumed engine off and engine restart felt as a duration of around 10 seconds in duration. I don't believe the altitude intentionally lost during the whole recovery process was greater than 1;000 feet. Once back in straight and level flight; I began assessing the situation; monitored engine and electrical indications; checked fuel indications and settings; finding no anomalies; no indications of malfunction; no improper configurations. Convinced that the engine had shut off due to air starvation / improper air/fuel mixture setting; I continued to fly the plane; using various levels of throttle; listening; monitoring the engine performance and other indications; not finding any indications of an unsafe condition of the aircraft; and believing the aircraft to be in safe working condition. I continued performing a few other maneuvers; this time properly leaned to the altitude and the engine running noticeable better.I returned to ZZZ airport as planned and uneventfully. I did not advise ATC or distress using radio or transponder; or requested any assistance throughout the flight; as I did not felt was necessary at the moment given the initial altitude and the following normal operating indications. I was aware of the landing alternatives near my location at all times during and after the event; but never made the decision to use them. I was sufficiently rested; sober from all substances; in a good emotional and health state; with no particular pressures or obvious factors that could have negatively contributed to the event.I believe the incident is a direct result of lack of knowledge on proper use of air/fuel mixture for high altitude operations; combined with unclear/partial information and direction from flight school instructors in regard to using full mixture for maneuvers. Instructors have in numerous times it is the recommended procedure as per flight school policy. It's important to note that I have never been given a document stating this information; but have only been told personally during flight instruction. Haven't closely studied the POH in its entirety; and the information regarding recommended leaning over 3;000 MSL; I was not aware of the proper setting as per manufacturer; and relied instead on wrong information for this particular scenario. School personal had assumed students would not use high altitudes for these maneuvers; and I was never asked during the dispatch process of the aircraft about my planned altitude for my maneuvers.Once back at the school; I returned the aircraft documents and keys; had the times logged; and communicated about the engine 'quitting on me' during the flight. After brief questioning; the person also came to the conclusion that the engine was turned off by accident and was not due to malfunction. Unknown to me if the person reported the situation. Later that night; not knowing if the event was reported; I messaged the person in charge of my schedule and class; this time having the expected response to initiate a reporting process. Unknown if the aircraft was inspected before the next flight the following day.The school immediate halted solo flights from being scheduled to me; and I am still under that condition for at least the remaining of the commercial portion of the program. The reason given for this is not particularly the involuntary engine air starvation; but the decision making leading into; and after the event. I have acknowledged these mistakes and believe to have learned from the experience. Since then I have informed myself further on the aircraft configurations and systems; understood the importance of reacting with precaution; expecting a worst situation than the perceived.The school has not specifically given any additional instruction specifically addressing the concerns from this event; nor have provided opportunities to practice and develop further my decision making and procedures. The limitation on only flying with instructors has only created scenarios where I am in great part limited to follow instructions. I currently feel lacking proficiency on solo and cross country skills given the lack of opportunity execute and make decision on these. My flights has been reduced to simulated instrument and occasional closely monitored and controlled local flying.During this time where I have exclusively flown with instructors; I have witnessed unsafe practices such as flying inside clouds under VFR rules with me using a view limiting device and unable to properly identify the situation; an instructor pulling the AHRS circuit breaker (in flight; on an instrument approach) against school policy to simulate partial panel scenarios; an instructor falling asleep during cruise; an instructor mistaking left pattern/right pattern; going against of what was instructed by Tower ATC; several instructors teaching to use the GPS and a single VOR to performs VOR checks (using GPS in obs mode simulating dual vor; or using GPS waypoints instead of ground references). These situations feel unsafe and are creating a growing concern over safety.