Short Circuits vs. Ground faults
A short circuit is what happens when a hot wire touches a neutral, ground, or out-of-phase wire. Normally, all hot wires are fused (or on circuit breakers), so the current flowing through the hot wire should cause the fuse to trip.
But, if the resistance is high enough, you can have a lot of current running though a wire that will NOT trip the fuse. GFCIs (aka RCDs) are for Ground Faults. They open the circuit when there is an imbalance in current flowing as depicted below.
Note that in the diagram above, if the fault occurs before the GFCI the GFCI will not interrupt the circuit.
How does a ground wire keep you safe?
If the hot wire comes lose and makes contact with a grounded surface it will short-circuit and trip the circuit protection. If it weren't there, you could be it's path to ground. Note that the "Earth" has a lot of resistance, so the ground wire is also bonded to the neutral wire (usually at the load center). Metal appliances (e.g., toasters) have the metal shell bound to ground for the same reason - if your toaster doesn't have a ground wire and the hot wire touches the shell, you become the ground for it.
Why doesn't my GFCI tester trip the GFCI?
Defective or wired backwards.
ref If the tester creates the imbalance by resistance-shorting to ground, and there is no ground wire
ref
So can somebody please answer my question as to what works to ensure my inverter trips if a little hand sticks a fork in the outlet?
That's what I got from my internet searches anyway. Seems like your best bet is an RFD, a good ground, and keep the breaker amps as low as possible. Hope that helps.