Agree with a lot of what has been said. It’s a wear-and-tear or facility-use fee that’s factored into board.
Some places do charge more for using a school horse. So in order of increasing expense, it goes boarder lesson on own horse > school horse lesson > haul-in.
It’s interesting to me that schoolhorse use would cost less than a haul-in. Wear-and-tear on a horse seems more significant to me than wear-and-tear on the arena, plus other people can still use the arena during a haul-in lesson whereas use of a schoolhorse is exclusive.
The place where I take jumping lessons charges the same for a haul-in or for the use of a schoolhorse ($10 extra). It surprises me that the schoolhorse surcharge is so low, especially because they don’t use their horses more than once a day. I think $10 is a steal for the fun horse I jump who has already taught me a lot. Haul-in fees are $10-15 around here so when I do start taking my own horse, $10 will seem fair too.