I can see how they would say that, but I agree it does matter with the horse's personality/calmness.
My seven month old is one who has the personality of a big labrador. There were times as a baby, and even now, where if you go in his stall he won't get up and will let you rub on him laying down, and will lay his head in your lap and nuzzle you back.
I have never had another one with a personality like his. He is friendly like a spoiled horse, without the disrespect and unresponsiveness.
I have had one broodmare that let me once visit her and pet her while she was lying down. She is older, and I thought she was dead. I got within ten steps of her and talked to her and whistles at her and she didn't flick an ear. When I turned to go break the news to mom and dad, she sat up. I guess she sensed my mood. I was just so relieved that she wasn't dead(I'm not a big fan of death), and she let me bring her an armful of the new hay, and rub all over her face while she was down.
I would say don't be discouraged if your horse doesn't let you. It takes a certain individual.
I do think trust comes into it a bit, but like the others said it does depend on the horses character aswell.
When a horse lies down it is at its most vunurable, which is why I think it takes trust. But then some horses might be more naturally trusting or accepting of whats around them, while a more timid horse would be much lesser.
Well, I had one of those moments where I thought my gelding was a goner! He was sprawled out in the dirt, enjoying the warm sun, I suppose, and would not get up when I called to him as he usually does. As I got closer, I could see he had one eye open, watching me. I walked up to him, he stayed down and let me love all over him. Very sweet! It was like having a BIG puppy!