If the course is set up and ‘judged’ as an xc course, we used to say, “over, under or through” the flags, as long as the rider gets between the red and green in one piece without falling off, running out or refusing.
The fence judge’s job is to watch each horse carefully to make sure they clear the obstacle without a refusal. Back in the day when I did this, a stop and a pop was ok, but if the horse stepped back from a stop with even one front hoof, it was counted as a refusal. It is important to make sure you check off each competitor’s number as they go through so if there is a stop, you will make sure it is attributed to the correct rider.
You should also become familiar withthe penalty zone around the fence (if there is one in a hunter trial), where the rider cannot cross his path, refuse or fall off (elimination). You should also make sure you sit where you can clearly see the fence, but you will be safely out of the way if a horse does refuse or try to run out, and in a place where you won’t scare or interfere in any way with the rider. You cannot cluck, offer advice or help in any way to get the horse over the fence.
I haven’t fence judged for a long time, but this is what I recall from my fence judging days. Perhaps someone who is more up to date can clarify this for you. It’s really fun. Dress warm!