The following is an opinion and only an opinion. It should also be understood that the person writing it has no clue about electricity or life the world and everything (42). If you use this information and electrocute yourself I am in no way responsible... but I would like to have your stuff!
I am really bad at explaining but let me try as far as I understand it. When you are talking about 120VAC wiring you have 3 holes in a typical outlet. If you are looking right at a wall outlet assuming the big round one is on the bottom, that big round one is ground, the small one on the right is hot and the larger one on the left is neutral.
Forget about the ground for now. In a perfect world there is a balance between the hot and neutral. If you have 1 amp flowing from the hot, you will have exactly 1 amp flowing to the neutral (ok, not exactly but for ease of use lets go with that). If the amount is not equal it is assumed that the current is going somewhere else, like ground or YOU. The GFI watches for this condition and will disconnect power as soon as it sees an imbalance.
In the case where there is a DC wall transformer, there is circuitry inside the wall transformer that will typically reduce the voltage using a transformer then rectify that smaller voltage using diodes and capacitors as well as lock it to a fixed voltage, like 12VDC using a regulator device. The transformer that reduces the voltage and the other circuitry will separate or isolate the DC voltage from the AC voltage. In that case, the imbalance occurs on the other side (the DC side) of the transformer and the GFI does not see it. There can be exceptions to this but it comes down to the design of the transformer and other items that you might have in the tank.
As far as the grounding probe, there is much debate as to the effectiveness of having one in your system, at least as far as stray currents go. In regard to using one with a GFI, I can't see how you could go wrong. When you have a ground probe in your tank you will have a direct path to ground which will enhance the functionality of the GFI. The problem is, at least as far as I have experienced, is false tripping of the GFI when you have power brownouts, flickers or just a device that leaks a bit of current. If you have your critical systems on a GFI and it trips when you are not around to reset it, you could have some real issues.
Faz