Terrific question. You had it right in your first sentence: “the same amount of energy must have been released during the Earth's history,” but then it gets a little mixed up when you look at various energies, some of which aren’t related to the question at hand (for example, the current internal energy contributes positive mass-energy to the Earth, rather than counting toward the negative binding energy).
Bear in mind that the gravitational binding energy represents the total potential energy required to separate all of the matter of the Earth out to infinity against the attraction of gravity. Playing that movie backward, we see that the gravitational binding energy has been radiating into space since before the Earth began to coalesce from the primordial dust cloud of the nascent solar system (and ultimately, all the way back to the end of the inflationary period). So a full accounting of that energy loss would be a long and difficult calculation. But we don’t need to do that calculation, which would require looking at even the previous generations of stars before the Sun, because we can derive it from the gravitational field strength we measure, and our best approximations of the Earth’s matter content and distribution.
As matter accrues into a gravitationally bound region, and eventually forms a solid object, it acquires kinetic energy in exact proportion to the potential energy lost to its new position deeper inside the gravity well. If none of the matter collided, and all the particles simply orbited the center of gravity, the kinetic energy of the masses would exactly balance their loss of potential energy to the gravity well, and the total energy of the system would remain unchanged by the presence of the gravitational field (this is just another way of describing a conservative field). But matter does collide, heat, and radiate some energy away as it loses kinetic energy. That radiated energy is the gravitational binding energy that we measure and calculate – it accounts for the mass that’s missing, aka the “mass defect.”
The internal energies of spin and internal heat all count toward the positive mass of the Earth, by the mass-energy equivalence relation.
So the answer is that the sources you’ve cited simply aren’t going back far enough in time to account for all of the energy radiated into space by the Earth’s matter as it coalesced from “infinity.” A thorough derivation of the gravitational binding energy can be seen here.
