What exactly are the 2 points where a magnetic needle becomes horizontal?

The key search word for this is "magnetic inclination", also called "magnetic dip", i.e., the angle between the local geomagnetic field and the local horizontal plane. There are exactly two points at which the magnetic inclination equals ±90°, but for any other angle (including 0°), there is a continuous curve of locations with that magnetic inclination.

Here is a map of the magnetic dip angle as of 2015:

^{Image source: Wikipedia}

The green line, at 0° magnetic inclination, is known as the "aclinic", and it is the locus of points with horizontal magnetic fields, and it encircles the globe. It is roughly parallel to the equator, and it never deviates more than 15° of latitude from it. It crosses the equator exactly twice: once off the mouth of the Amazon in northern Brazil, and once near Kiribati, roughly south of Hawaii.

These probably drift over time, due to the general evolution of the geomagnetic field. The overall shape of the 0° "isocline" (the green line) probably does not shift nearly as much as the location of the magnetic poles. The precise intersections with the geographic equator might move more substantially, but only because of the low angle of intersection between the isocline and the equator (which implies that if the isocline moves a small distance parallel to itself, the intersection will move laterally by several times that distance).