Play with timeanddate.com's times. It gives the total amount of civil twilight for the day (which is split in half). For example at Atlanta (https://www.timeanddate.com/sun/usa/atlanta) there's 58 minutes per day at the summer solstice, 55 at the winter solstice, and 50 at the equinoxes. As you go poleward those numbers increase but it's always summer solstice > winter solstice > equinox. This is basically down to the angle between the sun's path and the horizon and I think I get why that would be different in summer than in winter but I can't get my head around the fall/spring bit.

You'd be able to do it based on the math in the article. Civil twilight lasts from sunset until the Sun is 6° below the horizon. So in the spherical law of cosines in the article, you'd just set a to be -6° instead of 0.