How do I use the oval screw holes in drawer sliders?
Use the ones that suit you best. Drawer slides typically have more holes than you need because they need to adapt to lots of different situations.
My guess is that they are used to be able to adjust the direction vertically/horizontally so that I can adjust the position.
I agree. Drawer slides often come with a sheet of installation instructions, and all the ones I can remember seeing have called the oval holes adjustment holes or similar. Here's an article from Wood Magazine that says:
Many slides come with multiple holes and slots to allow fine-tuning
drawer position and accommodate a variety of mounting options...
I don't think that the oval holes are really meant to accommodate seasonal movement. Most of the solid wood drawer sides that I've ever seen have the grain running parallel to the length of the side, and while wood does move a little bit radially and tangentially, it really doesn't change much in the direction of the grain.
However, when both vertical and horizontal oval holes are available, which one do I pick? I can't use both at the same time, so I have to pick one dimension I can 'adjust'. How do I decide which dimension to pick?
Again, use the ones that make sense for your situation. As long as the drawer fits and slides smoothly, nobody's going to say that you did it wrong. For example, if you want a bit of adjustment both horizontally and vertically, you could use the horizontal holes on the case sides and the vertical holes on the drawer sides. If you then find that you're happy with the horizontal position but need more vertical adjustment than you've got, drive some screws into the vertical slots on the case side and remove the horizontal ones. Or, if you're using a jig to install the slides and have a lot of confidence that the jig will get the position right, use the round holes and skip the oval ones entirely.
Also, are these designed so that I use every hole in the hardware to attach, or can I skip the vertical if I use the horizontal?
No, you don't need screws in every hole. I'd install one screw for each set of holes. For example, the slide in the image you provided has three groups of holes on the drawer side, with each group having a round hole and a horizontal and vertical oval hole, so I'd install one screw in each of those groups.