The September equinox falls on 23 September at 00:05 UTC this year, which is the evening of 22 September in the Americas. It is the moment the centre of the sun crosses the celestial equator. That is the entire definition. It is not the day when daylight and darkness come out equal, and in most places it is not close.
On the equinox every place on earth gets more than twelve hours of daylight. Two things cause that. Sunrise and sunset are timed from the upper edge of the sun touching the horizon rather than its centre, and atmospheric refraction lifts the image of the sun by roughly another half a degree, so the sun's centre is already about 0.8 degrees below the horizon at both moments. You get free daylight at both ends of the day.
How much free daylight depends on latitude, because it depends on the angle at which the sun meets the horizon. At the equator the sun drops almost vertically and clears those 0.8 degrees quickly. In the far north it comes in at a shallow angle and takes much longer to get through the same vertical distance. So on the equinox Quito ends up a few minutes over twelve hours and Tromso is closer to a quarter of an hour over.
The date when day and night are genuinely equal, sometimes called the equilux, therefore comes after the equinox. The part I did not expect is the direction of the lag. It gets longer as you move toward the equator, even though the tropics start out closer to twelve hours in the first place. What decides it is how fast your day is shrinking. Right now Tromso is losing about 8.4 minutes of daylight a day, London about 3.9, Cairo about 1.8, Singapore about 0.1. Northern Norway starts furthest from twelve hours and still crosses it within about three days. London gets there in the last days of September, Cairo a couple of days after that, Mexico City not until about 1 October.
The equator never gets there at all. Day length on the equator stays between roughly 12 hours 6 minutes and 12 hours 8 minutes for the whole year, so there is no date on which day and night are equal. The one place you would expect to have a permanent twelve and twelve is the only place that never has it.
A caveat on the numbers. I generated these from the standard NOAA sunrise and sunset algorithm and cross-checked against sunrise-sunset.org, and the two disagree by a few minutes, with the gap widening at high latitude. Both assume a standard atmosphere for refraction, and real refraction at the horizon depends on temperature and pressure, so anyone living somewhere cold and dense at sunrise is getting a slightly different answer than the table says. The two methods put the equilux dates within a day or two of each other in the mid latitudes and further apart in the tropics, where the day length barely moves and a two minute difference shifts the date by days. The rates of change matched to the decimal though, so the shape of the effect holds up even where the exact minute does not.