I was going to say the same thing as Drum.
I would consider a "true" automatic horse to be one where any person can ride that horse, cue it wrong, and the horse is still going to run the same correct pattern.
But there's still a "decent" automatic horse that makes an honest run with very little help from the rider. This type of horse may not carry anyone through the barrel pattern flawlessly, but still needs very little help to do their job.
My ol' Beau was close to an automatic barrel horse. He was so honest
. Throughout his entire life, we knocked ONE
barrel. And that's because he somehow just brushed his nose on the 1st barrel during the turn. I'm still not sure how he did that. (And we woulda placed 2nd if he hadn't). Sometimes he'd anticipate a little on the barrel turn, and I had to make sure to push him past, but usually he got it right. He took a long time to learn things, but once he did, man did he stay consistent. (Pole bending on the other hand, he was anything but automatic ..... that little anticipater......)