Why is the summer solstice night shorter than the winter solstice day?
Aside: If you use your sunrise / sunset calculator you will see that day and night are not equal at the equinoxes. The reason for this disparity is the same as the reason for disparity between the length of shortest summer night versus the length of the shortest winter day.
The disparity is primarily a consequence of how sunrise and sunset times are defined and calculated. The calculated sunrise time is defined as the moment when the top of the Sun is expected to first appear above the flat horizon. Similarly, the calculated sunset time is defined as the moment when the top of the Sun is expected to finally sink below the flat horizon. The two key factors that lead to the noted disparity are that
- The Sun is not a point source. It appears to be a sphere with an apparent diameter of about 32 arc minutes. So even on an airless Earth, sunrise and sunset would occur when the center of the Sun is about 16 arc minutes below the horizon.
- The Earth's atmosphere typically (but not always) refracts light from celestial objects near the horizon. The troposphere usually has temperatures that decrease with altitude. This temperature gradient refracts light (and also sound; this is why you can sometimes hear far-away trains, but other times you can't). Most sites that calculate sunrise and sunset times use a fixed value of 34 arc minutes for refraction.
Adding the sun's apparent radius and typical refraction angle results in the center of the Sun being 50 arc minutes (0.833 degrees) below the horizon at the calculated times of sunrise and sunset. These two factors account for almost all of the observed disparity between the shortest summer night and the shortest winter day. Most sites do not go beyond the standard 50 arc minutes below the horizon calculation, so on those sites (which is the vast majority of sites that calculate sunrise / sunset times), those two factors alone account for all of the observed disparity.
User d_e made a great comment, which is that the effect depends strongly on latitude. At the equator the path of the center of the Sun at sunrise and sunset is always perpendicular to the horizon (ignoring atmospheric refraction). Even after accounting for the non-zero diameter of the Sun and atmospheric refraction, the path of the Sun is very close to vertical at sunrise and sunset. The 100 arc minute offset (2*50 arc minutes) amounts to a small time difference. Aside: The longest days at the equator occurs on the equinoxes rather than the solstices, which is when the days are shortest.
Looking at locations outside of the tropics and the polar regions, the longest and shortest days are always on the June and December solstices in the northern hemisphere, and vice versa in the southern hemisphere. Latitude has a significant effect as sunrise and sunset occur at increasing angles with respect to the vertical as absolute latitude increases.
As an extreme, consider Rovaniemi, Finland. This town is located just south of the polar circle. The most recent sunrise was at 1:31 AM on 6 June 2022. The next sunset won't be until 1:04 AM on 7 July 2022. That's nearly a 31 day long "day". A portion of the Sun remains just above the horizon even at local solar midnight during that 31 day long interval. On the other hand, a portion of the Sun will appear above the horizon on the December solstice for over 2 hours and 14 minutes.
There are multiple issues with this standard 50 arc minute offset. A lesser issue is that the Sun does not always appear to be 32 arc minutes across. The eccentricity of the Earth's orbit makes this apparent size vary.
A much bigger issue is the 34 arc minutes used for calculating atmospheric refraction. The 34 arc minutes used for atmospheric refraction is based on the average environmental lapse rate (how much temperature drops, on average, with increasing altitude) of 6.5 kelvins/kilometer. On a very nice day the lapse rate can be much closer to the dry adiabatic lapse rate of 9.8 kelvins/kilometer as opposed to average environment lapse rate. On a rather nasty day with a temperature inversion (lots of smog, fog, low clouds, etc.), the lapse rate can be close to zero, or even negative. The variation in lapse rate can make calculated sunrise and sunset times off by five minutes.
That said, there is no way to calculate sunrise/sunset times for anywhere on the Earth and for years into the past / into the future without using a canned value for atmospheric refraction (and for the Sun's apparent radius). So that's what those websites do.
Finally, if there's a mountain or tall building off to your east (or west), the sunrise and sunset time calculations do not account for that mountain or tall building. The websites that calculate sunrise / sunset times assume the Earth is a triaxial ellipsoid with no obstructions.