it would be a machine process capable of adapting any series of problems and datasets, further it could "learn" from doing said problems and evaluating datasets (language, math, DNA), and essentially reprogram itself to get exponentially better at "learning."

Take the 2+2=4 problem (and more broadly 2+2=5). AGI would be able to solve 2+2=4 without any additional information, separate from a dictionary, similar to how a child in preschool would learn math with blocks, similarly AGI would be able to think on the moral problem of 2+2=5.

We don't know what AGI really is because we have a technology limit, ie. we can't synthesize the human brain, beyond having a child. And we can't recreate the human brain organically because of the ethical implications (and well, we don't know how to either).