I notice the old fences around all have 2 horizontal rails for the
boards to attach to. I've also noticed that new specs often show
3 rails, without comment. I'm thinking that this might be due to
the fence boards being much thinner than they used to be.
I like to place my rails long side vertical, so I don't get
excessive sagging over time. But I'm thinking that unless I
expect the fence to see hard wind loads or other severe attacks
(kicks, etc.), 2 rails might be enough.
Any thoughts?

The purpose of the fence matters. If it is just for show, two rails is
enough. If it is to stop certain critters, three might work better.