What are the differences between a light probe, spherical image, LatLong, vertical cross, cubic (etc) versions of HDRs?
Depending on what application you are using, the names of the shapes of the HDR will change.
Most 3D programs want/need an image that looks like an unwrapped world map. This file can be called “Equirectangular”, “Spherical”, “LatLong”, and also a “Latitude Longitude” file. These are called different names, but they are all exactly the same!
Vertical Cross, Horizontal Cross, and Cubic HDRs… these are all just six 90-degree views of a scene or panorama. They just happened to be place together in a certain shape (vertical or horizontal cross) or saved out as six separate cube faces as six different files. Retouching is pretty easy with these as long as you do not try to retouch any of the edge pixels. If you need to do some cloning or color correction across the seams, then convert the image to a spherical image and work on that. Then convert back to a cubic format if needed.
“Light Probes” are full 360x180-degree spherical (or LatLong) HDR files that have been converted into a ball shape. Visually, it appears that they are chrome balls, but in reality they have a mathematically much simpler distortion and they may or may not have been created and captured that way. A few 3D programs directly support Light Probes (ball-looking) images, but almost all support LatLong shaped HDR images. When you make an HDR from a series of pictures of a chrome ball, this too, is considered a “light probe”. In fact, any fully immersive HDR panorama can be considered a light probe, no matter what unwrapping format it is in. They are also sometimes called "Light Maps" as well.