The two posts on changing the seasons (here and here) resulted in a lot of interesting information in the comments and it seems like there is quite a geographical variation in how the seasons are demarcated, with the US possibly being an outlier in using the solstices.
Reader ahcuah is a kindred soul and has kindly sent me the data he collected over a full year of the daytime high and low temperatures. He lives fairly close to Cleveland and so the data is similar to what I would have gotten. In general, the shape of the graph and the locations of peaks and the valleys should be the same over the entire northern hemisphere (I think) and inverted for the southern, so the pattern he gets is of far greater general utility than just for his location.
The recorded temperatures are only to the nearest whole number and this causes a problem for figuring out what the coldest or warmest days are since there are many days around those two points that have the same temperature. For example, the coldest daytime high stays at 34oF from January 14 to January 24. One could make a reasonable estimate that the coldest day is at the midpoint, which would be January 19th, but ahcuah knows that there is a better way to check this.
He did a Fourier analysis of the entire set of data and found that it converged pretty rapidly, with six terms in the series being sufficient. Using these Fourier components, he was able to recalculate the daily temperatures to greater precision and found that the coldest day is actually January 19, agreeing with the rough estimate. The coldest recorded nighttime low stays at 19oF from January 10 through February 4, with January 22/23 being the midpoint. The Fourier analysis says that the coldest night is January 22, again agreeing with the rough estimate.
While we can understand why the coldest days occur in mid-January, a month after the shortest amount of daylight hours given by the winter solstice (the reason being that it takes some time for the Earth to cool down), no obvious explanation comes to my mind as to why the nighttime low lags behind the coldest daytime high by three days.
Similarly, the hottest recorded daytime high stays at 84oF from July 16 through July 27 and the Fourier analysis pins the hottest day at July 19. The highest recorded nighttime low stays at 64oF from July 20 to July 28, with the Fourier analysis giving the peak on July 24, lagging by five days. The hottest days occur about a month after the longest day at the spring solstice, again due to the time lag for the Earth to heat up.
In addition to the interesting puzzle of why there is a three day lag between the daytime and nighttime peaks, there is also issue of why the nighttime peak in winter is broader (26 days) than the daytime one (11 days) but in summer is narrower (9 days vs. 12 days).