It's true. Highly recommend getting a globe in your hands to convince yourself about stuff like this. Close one eye, pretend you're the sun and observe the effect the tilt has and how it has no effect on the equator (apart from changing the angle of the sun; interestingly the sun's path alternares between being in the north sky and being in the south sky).

You'll also be able to understand time zones and great circles (flight paths) better with a globe.

Ok thank you! I think I can see how I am wrong now and why it doesn't actually make a difference.

To be clear, the days and nights are still not exactly 12 hours because the Earth does not take exactly 24 hours to rotate.

It's true that the Earth does not usually take exactly 24 hours to rotate, it's only 24 hours on average over the course of the year. However, the difference is only ever off a few seconds at most. Those seconds pile up across the days around perihelion and aphelion amounting to several minutes cumulatively but that doesn't affect the length of the day, only the amount that solar noon differs from noon on the clock.

The reason that days are a few minutes longer than nights at the equator is due to atmospheric refraction which makes the sun visible a little before it actually has "risen" and similarly, keeps it visible a little after it has "set".

In Garissa, Kenya, which is almost exactly on the equator, the day length varies between 12 hours 6 minutes and 12 hours 9 minutes. So it is 12 hours if you are rounding to hours.

Pontianak in Indonesia is right on the equator (I'm seeing 1 minute 14 seconds south latitude at Wikipedia).

timeanddate.com gives (https://www.timeanddate.com/sun/indonesia/pontianak) variation between about 6.5 minutes and 7.5 minutes above 12 hours, long at the solstices and short at the equinox. The difference is due to the sun setting vertically at the equinoxes and at a slight angle to vertical at the sunset.