In my 61 years of being involved with horses never once have I "laid a horse down or trained one to lay down". I just never ever had reason to do it. Case in point #1) One morning after a few hours ride I cooled my mare out then turned her out. When I came into the house I looked out a window and saw she had laid down to roll and got all four of her legs stuck straight up into the panel pen rails. Casted big time she was. I rushed out and slowed as I got closer to her, talking to her in a calm voice the entire time. My Candy laid there like a rock and I climbed over her barrel into the area between her front legs and back legs to free up her four legs. All that trust with never having taught her to lay down by any means.
Another case in point on training: I bought a POA mare from a trainer and his partner for my grandkids. The trainer told me he had trained the mare to do a reining horse spin. Thankfully he told me that, 'cause I was riding her one day and unknowingly found the reining horse spin cue by accident. I would have dreaded her going into that spin with one of my grand kids on her.