You are correct, to provide hippotherapy, a licensed PT, OT, or speech-language pathologist must be present. Hippotherapy focuses on the actual physical, occupational, or speech therapy goals. Riding skills are not a focus- typically the client will ride with a bareback pad or surcingle and does not control the horse. The rider may ride in different positions, like backwards, prone (face down) or supine (on their back), etc. Check out the AHA website for more info on hippotherapy: American Hippotherapy Association, Inc.
Conversely, in therapeutic riding, the participant focuses on riding skills (although physical or educational objectives may also be emphasized). Riders learn to control the horse as independently as possible.
To be a therapeutic riding instructor you *should* be certified. The largest organization for certification in the US is PATH International: PATH International
Some other organizations, like the PA Council on Therapeutic Horsemanship and CHA also offer instructor certification for teaching riders with disabilities.
In reality, anyone could go out and start teaching riders with disabilities. Obviously working with a center affiliated with PATH or at least with an instructor with some sort of certification would be the best way to go!
The term equine-assisted therapy is often misused, but should only be used to refer to therapy (in which an actual licensed therapist is present). This could include hippotherapy or equine-assisted psychotherapy (also called equine-facilitated psychotherapy, and requires a licensed mental health professional).
Check out the EAGALA website if you want to learn more about EFP: htpp://www.eagala.org