Terminology is kind of splitting hairs, but as someone who has experience (I was certified by government Animal Control in Northern California back in the early 00's and am currently a certified foster for the Humane Society), I think of it this way:
"Rescue" is an organizational term, usually implying 501(c)3 status -- a registered, non-profit organization that specializes in saving animals from imminent death. This includes severe, life-threatening physical malady OR those unwanted horses (e.g., Premarin babies) that are sold to feedlots. They will also receive owner-turn-ins (animals that are no longer wanted) for the purpose of re-homing them. The point is: If possible, a rescuer will re-home the animal to make room for other animals in need of rescue.
I consider myself a rehabilitator ("rehabber"): an individual who physically rehabilitates animals that are ill, restoring them to health, with the purpose of re-homing them. I didn't train the horses that I rehabbed, but my job included instilling basic ground manners so that they could be handled for treatment. Once they were physically well, I either re-homed them myself (if I had purchased the horse myself); returned them to AC per my contract; or placed them with a 501(c)3 Rescue organization, which would train them and re-home them. Occasionally, I would adopt one of my rehabbed animals -- we call it a "foster failure." ;)
That said, I really don't care if an individual wants to say that they "rescued" their horse or not. It has no impact on me; but it tells me that the "rescuer" is someone who is compassionate and wants others to know it. Can their compassion be misguided? Sure, but that's not my business -- unless their animal becomes dangerous.