It may sound off the wall but I always keep a nice size squeeze bottle of generic saline "the kind for contacts" in my locker. I try to find the kind without cleansers...sometimes you can find it, but I found it's very effective for washing out horsie wounds. I had a vet friend years back that told me to use that once for a bad cut on my Morgan's nose to get it cleaned up while waiting for the equine vet. I also like Vetericyn as it is a great germ/microbe killer that's great for everything from cuts to rainrot. Vetwrap, gauze, pocket knife, wire cutters. I think those are pretty much my mainstays right there.
Yes, you should always have your own basic first aid supplies. I am guessing their kit is for their horses and for emergencies.
You need to be prepared for your horse.
Lots of good things have been listed already.
I agree on the saline. I keep a bottle of plain saline solution. It is great for eye issues and spraying in things if a hose is not available.
Be careful spraying into large wounds. You can wash the debris in further. If it is the type of would that requires a vet let the vet do the debris removal.
Also have a thermometer so you can take temperatures. A list of horse vital signs so you will know what normal is. If the list includes a quick 'how to' it will give you help when you are in a panic. You should practice checking these things now, when the horse is healthy.