Grain = feed, typically. However, I prefer to use the word “feed” because not every feed actually contains grain. There are plenty of pelleted horse feeds that contain no grain, and it annoys me when people call them “grain.” But, they’re basically interchangeable in most people’s vocabulary. I don’t personally know anyone who calls hay cubes or hay pellets “feed,” although that is technically what they are.
The “worming” thing annoys me too. You de-wormed your horse, you didn’t worm it.
Barn = stable = stables
Different regions have “standards” for their words, you’ll get used to it. I have one boarder who calls a flake of hay a “pad.” Literally never heard that one before.
The coach/trainer/instructor one is a bit more convoluted, and wildly discipline specific. “Instructor” to me is usually someone who predominantly teaches lessons, doesn’t do much horse training. A “trainer” usually mostly trains horses, but may also teach some lessons, and is usually more involved in care decisions too. “Coach” is mostly an easier term for instructor, at least for me. My dressage trainer is a trainer, she trains horses and teaches some lessons. I usually call her my coach, because I just take lessons from her, she’s not ridden my mare. I do have her do some training rides on a gelding I have in for a friend. So in regards to the gelding, I call her my trainer, even though I am still in charge of his feed/care decisions. In regards to my mare, I call her my coach.
But a racehorse trainer doesn’t ride the horse, the jockeys/exercise riders do. This trainer is usually in charge of feed decisions and exercise regimen, but doesn’t actually do the exercise.