OP, your location, what kind of weather you have, might be helpful to us for ideas to suggest. I will certainly second the Native Species if possible, they are made to survive in your location.
I have Vinca (Periwinkle) vine beside the house, just covers the dirt nicely, survives under dogs laying on it and in my location, stays contained. But never read up on it in relation to horses or how deep the roots go. Mine is leafy, evergreen year around even under the snow.
Avoid Chameleon Plant, that is a terrible thing to get going in your place, though very colorful. Spreads like wildfire and almost impossible to remove since every part or piece can resprout if you miss removing it from the dirt. I can’t believe some of the terrible invasives allowed to be sold by local Nurseries! No wonder it is so hard to control in the wild areas they spread to!
You might check out the short growth decorative grasses. They stay short, have good roots, only need trimming off once a year. Most don’t spread either, clump forming, so not problem plants. Lots of selections in heights of 12" or less, attractive colors like red, yellow or blue or even some black leaves.
