It does track your horse’s steps as you ride. I don’t know accurate it is in terms of how many steps per stride it is logging since I am not that crazy about knowing the exact number of steps I take in a day so I haven’t paid that much attention.
I haven’t found anyway of shutting off the step tracker. As far as I can tell it is built in to the system and can’t be turned off. The only way to do it is if you are in a wheelchair, there is a base setting so that it tracks wheelchair pushes with your arms rather than steps. I’ve just accepted that it is one of the flaws of tracking your riding with the Apple Watch.
And the “equestrian sports” option for a workout has actually always existed, and is not new to the Series 3, in case anyone hasn’t realized. The new operating system included equestrian sports as an option for everyone once you update your watch. Even if you don’t want to update, there is still a way with the old system–select “other” as a workout option, and then at the end of your first ride, when you hit “save” select “name workout” which gives you a huge list of sports, including equestrian. It will stay in your workout list after that so you can just select it when you ride.