I would say both descriptions are lacking, but the second description of latitude and longitude (i.e., the "new" way) is maybe slightly better than the first (i.e., the "old" way), but the introduction of "horizontal" and "vertical" are problematic and imprecise. Neither description is wrong, but both are missing key elements. The way I would describe them are that latitude and longitude are a spherical coordinate system (i.e., a way of slicing up a spherical shell) allowing you to uniquely define a position on the surface of the sphere with a set of two coordinates. In terms of the orientation of the lines of latitude or longitude, it's more correct (and maybe easier) to consider them with respect to the rotational axis and the equator of the planet. The equator is a line that marks the intersection of the surface of the planet with a plane perpendicular to the rotational axis that goes through the center of the planet. Lines of latitude (other than 0 degrees, i.e., the equator) are the intersection with the surface of the planet of planes that do not go through the center of the planet, but are parallel to the equator. For this reason, lines of latitude are sometimes referred to as "parallels". Line of longitude, or meridians, are the intersection with the surface of the planet of a series of planes that go through both the center of the planet and the axis of rotation. So if you were to cut the sphere along any line of longitude, you would divide the sphere into two equal hemispheres, but if you were to cut the sphere along any line of latitude but the equator, you would divide the sphere in two unequal halves. Then, in relating the coordinates to the lines of latitude and longitude, because we are on a sphere, we measure these in degrees. The angles between lines of longitude are a little easier to visualize since these all intersect at the poles. The angle between lines of latitude are a little more difficult to picture since by definition lines of latitude to not intersect. Here, it's useful to consider a hemispherical slice along a meridian, i.e., we slice the sphere in half perpendicular to the equatorial plane, bisecting the poles. Then, on this hemispherical slice if we draw two radii between any two lines of latitude and the center of the sphere, the angle between those two radii is the angle between the two lines of latitude. The final piece of the puzzle is the origin of the coordinate system. For latitude, this is not arbitrary and it's fixed to the equator, i.e., to define the degree of latitude we measure the angle between the equator and that line of latitude per above. For longitude, it is arbitrary and we have to pick some meridian from which to measure our angles. Once we have a set of lines, ways of measuring angular distance between those lines, and origins for the two sets of lines, we now have all the pieces to define a unique coordinate for any place on the surface of the sphere. More on reddit.com

Here is a Visual Chart for you More on reddit.com

Point at a spot on google maps, the coordinates will appear. More on reddit.com

That is correct, but keep in mind that in this case, minutes are 1/60th of a degree, and seconds are 1/60th of a minute. More on reddit.com