Grounding issues and ground tester
Those plug in testers can be total crap. I have about a dozen of them and 2/3rd of them show good circuits while the other 1/3rd show problems, and not all the same problem. 

You might be surprised to find out with a high impedance meters ( the type most meters are) you can plug the meter lead in the hot terminal (the smaller one) you may read a voltage very close to your supply. 

As for the neutrals and grounds being in the same bar this correct at the main panel but every place else it is not allowed. Since you just replaced the water heater and are having problems I would add a ground bond to the outlet pipe, animals are more sensitive to leakage currents than we are and this might solve the issue. I would also suggest to drive a 2nd ground rod. 

Standard rods are only 8' long in the past when I have used pipe I use a full stick that is usually 10' long. Current code requires 2 if this is your only grounding system 6' or further apart. Unless the single rod has a 25 ohm or less resistance (takes a special meter to measure so most just drive 2 ).

Also, I was initially thinking we were discussing a water heater, now with the edit I see it is a "cal rod heater". Put your meter probe in the water and then to ground. If you measure a voltage the heater has leakage and is faulty, take it back. All my tanks are metal and sit on the ground outside and we only have a few weeks a year where we have to break ice. I would encourage you to put this heater on a GFCI outlet! But I believe you have a faulty heater. 

Even with no ground the hot conductor of the cal rod is in the center protected by a metal jacket. It is basically a big resistor that gets hot so either the jacket is cracked or the connection points are not properly sealed - this is the reason the horses are getting shocked. Not a grounding issue.