3 lamp tester, into the trash it goes.
Grounding systems are simple but critical. My philosophy is "nuke it from space, it's the only way to be sure". The 2014 Electrical Code agrees with me, it greatly liberalized the rules for retrofitting grounds.
I break grounding systems into 3 sections, from the, um, ground up.
The Grounding Electrode System
This is the ground rod or water pipe tie-in (or Ufer if your concrete pouring guy was a smart cookie). Modern standard for ground rods is two of them some distance apart, connected by copper wire to your service panel's chassis and its grounding bar. I leave nothing to chance, bonding to both the chassis and the grounding bar.
The Neutral-Ground Bond
In your case, the neutral bar and the ground bar are the same bar, which is the sincerest neutral-ground bond possble. These often tie to the panel chassis with a green "ground bonding screw", I have seen cases where this screw was burned up by a prior overload.
The Equipment Grounding Conductor (EGC)
This is the "ground wire" that should be present in every circuit from the ground bar to the point-of-use. Because people understand the importance of it, this isn't usually where the problem lies.
I assume you checked this on the wiring to your horse trough, so there is a real possibility that the problem isn't this ground exactly, but all grounds in your house. The horses just discovered it first.
Ground fault detectors
GFCI "receptacles" can go right next to 3-lamp testers, as far as I'm concerned. Too many people buy a 10-pack at Home Depot and replace every receptacle with one, without a moment of thought, this creates a giant mess and a "Yo Dawg" joke. It also wastes money better spent on other safety apparatus. GFCI protection is a great concept and is important, but it is better installed intelligently, and only needs to exist at 1 point in a circuit. From there, it can protect everything downline of that point.
Since you have at least 1 circuit that runs around outside, yeah, GFCI protection is required for that, and in theory will stop horses from getting shocked. Put the GFCI device somewhere early in the circuit, so the unit is still indoors. Putting a GFCI device outside is just wasteful.
You can and should fit GFCI on this circuit, I recommend as a breaker or receptacle inside. However will simply convert it from a "horses getting shocked" problem to a "GFCI trips and refuses to reset" problem. The core problem will remain the same: your tank heater has a ground fault.
The only upside of opening up every receptacle to change it to GFCI is that it forces you to open up every receptacle and inspect it. There you may find all sorts of interesting problems left for you by history, and one or more might explain the problem you're having today. But just do that, then; no need to spend $20 per socket.
Or ground is different at the horse trough
A longshot possibility is that the tank heater is perfectly grounded, to the house, but the ground the horses are standing on is at a different potential than the house's ground rod. That could be caused by a significant amount of leakage from some other apparatus in the area. The earth has basically become a giant rheostat (open resistor) between two different points having differing voltage in the same system.
However for that to happen, somebody's electric meter would have to be spinning pretty quickly. And that would also be the case even if the horse trough heater's breaker was off. So powe off the heater and see if the horses still get shocked.