It's worth starting with the problem that lines of longitude - defined as great circles that pass through the rotational axis of the planet - solved. Specifically, in early navigation (primarily at sea), measuring latitude was relatively simple because either through observation of the sun position or star positions you could work out how far north or south you were of the equator. This is in part because that the equator is not an arbitrary line as it's defined as the great circle (the only line of latitude that is a great circle) 90 degrees from the rotational axis of the planet (and this is why it makes sense to define lines of latitude that represent fixed angular distances from the equator as what you were measuring was the angular distance from the equator and/or from the pole). In contrast, figuring out how far east or west you were while at sea was more challenging. The idea behind longitude was to define great circles where positions along a singe line of longitude reflect locations where the sun will be at its zenith at precisely the same time. Then, with precision time keeping, a reference point (the prime meridian), and the difference between your local zenith time and the zenith time at the prime meridian, you could work out how far west or east you were in angular distance. This would not work with lines of longitude that were parallel. If we imagine that now with more advanced ways of measuring positions there's nothing necessarily tying us to lines of longitude as originally defined, there are a few problems that pop up. One is again related to time. Specifically, from above, one property of a line of longitude is that it represents points along the surface of the Earth at the same time with respect to the position of the sun in the sky. A system with parallel lines of longitude would not maintain this property (except at whatever arbitrary position you defined the equator equivalent in this hypothetical longitude system). More importantly, if we imagine a system where both lines of longitude and latitude were internally parallel (and orthogonal to each other), we've introduced a non-uniqueness to coordinates. At present, latitude spans -90 to 90 degrees and longitude spans from -180 to 180 (or 0 to 360) so each point on the surface has a unique coordinate pair. However, if we had a system where longitude spans from -90 to 90, any coordinate would represent two possible locations. I.e. a coordinate of 0 latitude, 0 longitude could represent either of the two possible intersections (on opposite sides of the globe) of the latitude equator and longitude equator, and the same would be true for every intersection of lines of longitude and latitude. It probably goes without saying, non-uniqueness is a really bad property for any proposed coordinate system. The arbitrary nature of the position of the "equator" and "poles" of this hypothetical version of longitude is also a bit of a problem, or at least, it introduces a pretty annoying set of properties to a coordinate system designed in this way. More on reddit.com

What is the difference between... ... Hint: Geographic, or Unprojected, refers to how we represent the Earth in spherical coordinates. "Longitudes' ' and "Latitudes" are the most popular systems for finding locations on the globe. Longitude and latitude are both angles measured using the earth's ... More on vedantu.com

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

Latitude is north-south and longitude is east-west, so latitude is analogous to Y (up-down) and longitude is analogous to X (right-left). More on reddit.com