AGI is the persistence of a technoscientific myth of contact with the superhuman at the heart of a secular enterprise; an allegory of faith in a disenchanted universe.
I don't think such an arbitrary definition is useful since it would lack specificity and reduce phenotypic plasticity to a set of supposedly human-exclusive measurable objectives. The boundaries of human beings are blurry. If you want to study a brittle abstraction go on with GI.
The irony is that being surrounded by so many intelligences: bees, whales, chimpanzees, DNA molecules, computers, beetles, slime mold and even the earth as a gaian ecosystem, we still feel the necessity to recreate the strangeness that we miss in a computational abstraction.
It seems to me the failure to recognize ourselves fuels our thirst for confirmation from trully 'general' superhuman intelligences. Thus, AGI stems from a failed concept of the human, which under a biological reading includes the inhuman since we are not only made of human cells.
So your argument is that there's no GI because GI can't be defined? Hence your statement 'there's no GI' has no truth value. You don't know if it halts.
I am taking a step back and questioning the necessity and reasonableness of giving such definition in the first place, not arguing against its possibility. Of course you can define it however you like, I just think it won't take you far for the reasons I have given.