However, if you do any serous riding that involves traveling a lot of miles you'll probably want to spend some time giving your horse a break by dismounting, loosening the cinch and walking them on a lead for a few miles during the day (or maybe not, depending on how hard you want to work your horse). If you do you'll be much happier with the comfortable walking foot gear that works equally well for riding vs riding boots..
Also, if/when you get your own horse (or a leased one), I think walking with them helps build trust & awareness of each other. I had to hand-walk my horse for about a year, as injury rehab, before I rode her more than a couple of times. (And, barring a couple of tourist-type trail rides, she's the only horse I've ridden.)
I like the Ariat Terrain boots, too. They're as comfortable as any hiking boots I've ever owned. The soles are maybe a little smooth for slick rock, but otherwise I'd have no problems hiking in them all day.