I've read that only only in the equinoxes do you get to see the sun rise in the exact east regardless where you might happen to be. However I think this is not absolutely true everywhere, but only in the equator, and in the south the Sun will rise just a tiny bit to the north, but this tiny bit to the north is negligible due to how far the Sun is from Earth. If this reasoning is correct, when would the 'true' equinox be for any country in the south? Can you provide an example and how you did it, please?
The celestial equator's plane intersects the horizon due east and due west, no matter where you are on Earth (except at the poles where the horizon is the celestial equator).
Those that have the sunrise at the exact time of the equinox will see the sun rise in the east (90º). For the rest, it will be too late, the sun will have made the crossing, and it could be something like 90.00001º or 89.99999º, although it will still be perceived as due east.
Sunrise or sunset is usually defined as the moment when the upper limb of the Sun appears on the horizon. This is conventionally computed as the time when the center of the Sun is 50' below the horizon, using a 16' angular radius and assuming 34' atmospheric refraction.
Let δ be the Sun's declination and φ be the observer's latitude north of the equator. Near an equinox, the Sun's azimuth at sunrise or sunset is approximately δ / cos φ + 50' tan φ north of due east or west.
Solar parallax changes δ by about -8" sin φ, shifting the sunrise/sunset azimuths about 8" tan φ southward. This is about 1/400 as large as 50' tan φ, so we can neglect it.
For example, an observer self-isolating on a yacht at 50°S, 145°W, has φ = -50°. Sunset on 2020-03-19 occurs within a few minutes of the equinox; δ = 0°, and the Sun disappears at azimuth 269.0°. At sunrise on 2020-03-21, δ = +0.59°, and the Sun first appears at azimuth 90.1°. The NOAA Solar Calculator may be useful for other cases.