Drace what you're describing there as "vertical reading" is basically how your brain's pattern matching works. You build up a collection of small patterns that your brain recognizes on instinct, and then you break more complicated patterns into collections of smaller ones, usually in multiple levels. Its also a way to do "clustering", as it's sometimes called.
Clustering is a memory technique. It's often mentioned that your brain can only remember 7 things at once, but everyone knows you can easily recall things like 10 digit phone numbers (or however many digits they are where everyone lives) even if they're longer than 7. How do you do it? You break it into sections. Here in Canada, a typical phone number might be something like (684) 721-8632. Your brain would break it into 3 sections: 684, 721, and 8632. Each of those is less than 7 digits each, and there are less than 7 of those clumps there. when you have a whole bunch of notes on screen, this has to get a bit more complicated, but the theory is the same.
You break large patterns into smaller ones, and those smaller ones into even smaller ones over and over. This is all totally subconscious, but as far as we know this is basically how your brain actually processes things.
I think the reason people tend to read "horizontally" is because when you have so many things to try to pay attention to at once, you try to decide how to use your attention, and I think most people try to consciously read each "horizontal pattern" (all the notes that happen at the same time) and try to ignore the other notes around it.
The way I read is that as notes pass through the area I concentrate on, I build a mental picture of the entire pattern and identify all the "sub-patterns" while the notes are moving fro the area I focus on towards the judgement bar. By building a mental image I can identify the patterns, compare them with the music and use that to predict the upcoming patterns. It's a bit hard to describe exactly.