I think there are two different things that are getting mixed up here and both are valid. The rule of thumb of 1 hour for every 10 degrees Fahrenheit takes into account the fact that algae growth and chlorine consumption generally is slower at colder temps and faster at warmer temps so it's more important to prevent "dead spots" of chlorine (when the pump is off) when the water is warmer as the chlorine will get used up faster with the warmer temps. The rule of thumb of one turnover per day takes into account the size of the pool and the pump flow rate. Obviously, one hour with a slow pump on a large pool will not circulate nearly as much of the pool water as a fast pump on a small pool. However, the turnover rule doesn't take into account temperature variation. In the winter with pool water in the 40's, very little circulation is needed. But as was pointed out, if freezing is a possibility, then some run the pump while others drain their pipes and let their pools freeze over. In the winter, I have found that I can run my pump only 2 hours per day which is equivalent to about a 40% of a turnover (my pump is oversized and I'm going to get it replaced with an Intelliflo variable speed pump soon) and that's probably more than needed at pools temps of 45-50F.
In a perfectly ideal world, you would have a variable or low speed pump running 24 hours a day in a pool with perfect circulation (no corners) and turning over the water once during "typical" pool water temps of 85-90 and light bather loads. But we don't live in such a world since our pools don't have perfect circulation and we have solar panels that require faster flow rates to operate efficiently and most pumps are fixed speed. So generally, pumps are run long enough to prevent cloudiness since filtration of suspended particles is one of the main purposes of running the pump (the other is to circulate the water so that chemicals, especially chlorine, are evenly distributed). No rule of thumb is going to be perfect for every pool but the "one turnover per 24 hours" rule makes sense in terms of the physical chemistry while "less than one turnover at cooler temperatures" also makes sense and "more than one turnover per 24 hours" is appropriate at higher bather loads or if lots of organic junk (e.g. pollen) gets into the pool.
For good circulation, it is more important to have the pump running over more of the 24-hour period of time and neither rule accounts for that. For proper filtration to keep the water clear, that is a function of turnover rate though what makes the water cloudy to begin with may be a function of temperature. So you're both right, but there's not a simple answer except that if your water is cloudy or gets algae in corners and running your pump longer fixes that, then you know you've got the minimum amount of pump time needed.
