Flight instructor reported their student solo flying a light sport aircraft misread their fuel system and the engine quit while in the pattern. The student landed the aircraft.

Narrative

During the preflight inspection my student and I visually checked the plane's fuel level and observed that it required refueling. I brought over a fuel truck and refueled both wings so I could see at least 2 inches worth of fuel through the fuel port. Student and I confirmed and agreed that the amount we visually had would be more than sufficient for our 1.5 hour flight and about the maximum we could take with our weight and balance. The previous time this plane was flown was the night before by student and another instructor. The binder did not show any current Squawks or concerns for this aircraft. With both student's and my weights combined; we could only carry about 12 gallons of maximum fuel to not be over max gross weight. With the extra hot temperatures and low-pressure weather; I was extra considerate of this factor prior to our flight. And as a reminder; we do not have an exact way to identify the amount in the tanks; it is all estimations. We are taught that if you visually verify have fuel at the bottom of the tank; this equates to about 7 gallons of fuel. This being said; adding an extra 2 inches of fuel is equivalent to about 7-10 gallons of fuel per wing. This consideration would give us enough fuel for about 3 hours of flight (assuming the 5-gallon per hour fuel flow) more than double what we needed.We got in the plane and went through our preflight checklist. At this time we noticed that the fuel gauge was reading lower than what we had just visually confirmed. In taxiing the plane from the parking spot to the fuel pit area and doing a 180 turn; the fuel quantities fluctuated rapidly; making us skeptical of an accurate reading on the fuel gauge. I re-cycled the EFIS and EFIS backup to see if there was a glitch with the indication on the gauges. After re-cycling the indication on the fuel gauges and the levels still remaining the same; we felt confident in the amount of fuel we saw visually in the tanks and agreed to continue to take off. Student and I did about 3 laps in the pattern together. During this time; there was no indication of any issue other than the fuel gauges still fluctuating to the same amounts it was while on the ground. After 3 laps and perfect landings by student; I felt confident in student's ability and readiness to solo; so we taxied back and I got out of the plane and instructed him to do the full run-up and checklist. I stood by on the ramp with a walkie-talkie to hear communications and record his first solo.Student proceeded on his own and after completing another run up he did 2 laps in the pattern with full stops and taxi backs. He then began his third lap; and after turning downwind I heard on the communications that he reported an engine failure. I tried to respond or hear the tower response on the walkie; however; he was holding down the push-to-talk button and I could not communicate. I watched as student was able to land safely back down on [Runway] XXL and requested a tow. As soon as he landed I confirmed that he was safe and ok. He then described the details of what happened in the air. He said that upon turning downwind; he experienced engine sputtering and eventual engine shutdown. He attempted to restart the engine but ultimately prioritized landing back on the opposing runway in time. In discussing further details during our debrief; he mentioned he did not switch to the fullest tank on the checklist during the run up. My suggestion for the future would be to always confirm any misreadings with maintenance before continuing to fly; or get a third set of eyes on the levels. I would also make a photo record of gauges before and after flight for confirmation. I would also suggest a more detailed fueling course or run down as a CFI to ensure consistent standards across the board. The fuel truck training consists almost entirely of runway incursions and who passes can be given out to; rather than refueling details such as fuel tank anatomy and behavior. In ground school; wetalk about fuel systems but not the unique nature of the aircraft gas tanks and refueling. Just like Club Students need to be checked out every 60 days; instructors should be required to have a similar recurring checkout on these practices.After discussions with management following the event; I learned that if the tanks are more low than usual when you fuel to a level that you are used to seeing; you may come back a few minutes later to see the fuel has settled more to the root of the wing. This causes a delay in the visual appearance of fuel levels. This is not mentioned in the POH. Having this knowledge would have changed my decision-making process for this flight. I followed what I was taught by my instructor during my time as a student. I have also heard from numerous other CFI's that the fuel gauges have a high level of inaccuracy and to always visually confirm the fuel levels before flight. I did that during this flight and should have received a final sign-off from maintenance after seeing a higher discrepancy on the gauges before continuing on; to err on the side of caution. Also when soloing students in the future; after exiting the aircraft I will refuel to the new maximum allowable fuel amount for the new weight and balance without my weight. I would like to also confirm that this airplane reaches the certification that the gauges are accurate at zero. I also hope this instance can help reinforce fueling standards and for others to reconfirm if any instruments are in question.