Dear B&B: On the other side of the country (PA) I recently had reason to try this (excavation on new property for indoor/outdoor ring driveway, etc.). We got all our summer rain in June (10") while the excavated ground was being worked, or rather, waiting for the rain to stop to finish the work and reseed. So I was stacking up straw bales to divert water to acceptable areas and keep mud away from problem areas.
My guess is that your slope is much steeper than mine. In the worst areas I had we put down #4 stone (that is big stone gravel, twice the size of a man’s fist), put in a silt fence (the tough stuff is reinforced on the down side with sturdy wire mesh) downhill from the stone, and put straw bales on each side of the fence. we let the water divert to one side/end of the fence and this pretty much worked. If you put gravel down (I mean, like a 20’ wide pad) it will slow the mud and water, although in a flood just gravel by itself the water will dig channels - but still not as bad as all dirt.
The stone gravel and silt fence is needed if you really get a flood, it will hold better than ground against water, and muddy water will try to find a way around it (you said you had a direction that it could be diverted…
Straw bales themselves will likely change the course of the water but if it’s going fast & hard the water may run under. For holding dirt but letting water flow I found open flakes of straw more helpful in some areas than the bales, arranged in kind of S curves going downhill (this was going down a paved driveway) so you might think of that as well.
I did have straw bales divert water on shallow slopes - the water was going more slowly and the dirt dropped out in the ‘S’ curves. I DID have to give the water a place to go, though, so was digging little trenches in the grass until it had enough drop in it to go down that line.
In the long run if you have any chance of fencing a buffer of vegetation above your neighbor, sloping toward the acceptable drainage location, that would be very helpful. You’d likely need to fence your horses off that buffer to keep the vegetation in good shape. I would suggest a gravel berm against the vegetation buffer to help against the fast floods.
FYI this is an amateur who hopes to never have to do this again (happily grass has grown where needed) and others with more experience in the field may have better ideas.