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