Actually both work, but for different reasons.
When shooting sports, I was taught to ALWAYS use BOTH eyes, to avoid being splattered as you are tracking the quarterback or receiver (American football). And basketball is almost as dangerous as American football. The vertical with right hand up will keep your left eye clear to see to the left.
But for safety we shot horizontal, so both eyes could see around us.
I was down on the sidelines or the court floor, so safety/survival was a concern.
Comment about the right side pix, the guy's hand is not on the lens. He can't track focus this way, so the position is not usable for action shots (like kids at a birthday party), as the focus is always changing. And for a medium or long lens he has to hold the lens with his left hand, not the left side of the body, for weight balance/support.
I generally shot SLRs right side up. I can support pretty steady with the left under the body with my fingers on the focus ring, or for long lenses under focus/zoom ring, and with the right I can work the film advance lever easier and faster. If I really had to get steady then I would go right side down, to tuck in better, but film advance was slower as I used my thumb for both the shutter release and film advance lever. When I shot with a manual flash, then it was right side down, as I had to use my left hand to hold the flash above my head.
I shoot my DSLR vertical with right side down, because I can tuck better for a steadier hold, and I don't have to worry about a film advance lever. But if the position feels better to go right side up, I will do that.
IOW I use whatever method/position that gives me the best results in the situation.
BTW, I do not use a motor drive, because of the added bulk and weight of an already heavy camera. And as I get older the weight becomes more of a factor. So I hand crank the film advance lever. If I want to shoot FAST, I will shoot my DSLR, where I also don't have to worry about the cost of shooting all that film in a few seconds.